Number Theory IV: Transcendental Numbers

A. N. Parshin I. R. Shafarevich (Eds.)

i‘

Number Theory IV Transcendental Numbers

Springer

i‘,

Preface This book was written over a period of more than six years. Several months after we finished our work, N. I. Fel’dman (the senior author of the book) died. All additions and corrections entered after his death were made by his coauthor. The assistance of many of our colleagues was invaluable during the writing of the book. They examined parts of the manuscript and suggested many improvements, made useful comments and corrected many errors. I much have pleasure in acknowledging our great indebtedness to them. Special thanks are due to A. B. Shidlovskii, V. G. Chirskii, A. I. Galochkin and 0. N. Vasilenko. I would like to express my gratitude to D. Bertrand and J. Wolfart for their help in the final stages of this book. Finally, I wish to thank Neal Koblitz for having translated this text into English. August

1997

Yu. V. Nesterenko

Transcendental N. I. Fel’dman

Numbers

and Yu. V. Nesterenko

Translated from the Russian by Neal Koblitz

Contents Notation

. . . . .. . . . . .. . . .. . . .. . . . .. . .. . . .. . .. . . .. . . .. . .. . . .. . .. .

Introduction 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Chapter

11

................................................... .................................. Preliminary Remarks Irrationality of Jz ..................................... The Number 7r ........................................ ............................... Transcendental Numbers .................... Approximation of Algebraic Numbers Transcendence Questions and Other Branches ..................................... of Number Theory The Basic Problems Studied in Transcendental ....................................... Number Theory ................... Different Ways of Giving the Numbers Methods .............................................

1. Approximation

of Algebraic

9

Numbers

..........

..................................... 51. Preliminaries 1.1. Parameters for Algebraic Numbers and Polynomials ..................... 1.2. Statement of the Problem ............ 1.3. Approximation of Rational Numbers 1.4. Continued Fractions .......................... ....................... 1.5. Quadratic Irrationalities ...... 1.6. Liouville’s Theorem and Liouville Numbers

11 11 13 14 15 16 17 19 20 . . . .. ..

22

... .. .. . . .. . .. . ..

22 22 22 23 24 25 26

.. .. .. .. .. ..

.. .. .. .. .. .. ..

2

Contents

1.7. Generalization of Liouville’s Theorem .................... 52. Approximations of Algebraic Numbers and Thue’s Equation ..... 2.1. Thue’s Equation ....................................... 2.2. TheCasen=2 ....................................... 2.3. TheCasen>3 ....................................... 53. Strengthening Liouville’s Theorem. First Version ofThue’sMethod .......................................... ................................ 3.1. AWaytoBoundqO-p ........ 3.2. Construction of Rational Approximations for $$ 3.3. Thue’s First Result .................................... 3.4. Effectiveness .......................................... 3.5. Effective Analogues of Theorem 1.6 ...................... 3.6. The First Effective Inequalities of Baker .................. 3.7. Effective Bounds on Linear Forms in Algebraic Numbers ... §4. Stronger and More General Versions of Liouville’s Theorem and Thue’s Theorem ....................................... 4.1. The Dirichlet Pigeonhole Principle ....................... 4.2. Thue’s Method in the General Case ...................... 4.3. Thue’s Theorem on Approximation of Algebraic Numbers . . 4.4. The Non-effectiveness of Thue’s Theorems ................ $5. Further Development of Thue’s Method ....................... 5.1. Siegel’s Theorem ...................................... 5.2. The Theorems of Dyson and Gel’fond .................... 5.3. Dyson’s Lemma ....................................... 5.4. Bombieri’s Theorem ................................... $6. Multidimensional Variants of the Thue-Siegel Method .......... 6.1. Preliminary Remarks .................................. 6.2. Siegel’s Theorem ...................................... 6.3. The Theorems of Schneider and Mahler .................. 57. Roth’sTheorem ........................................... 7.1. Statement of the Theorem .............................. 7.2. The Index of a Polynomial .............................. 7.3. Outline of the Proof of Roth’s Theorem .................. 7.4. Approximation of Algebraic Numbers by Algebraic Numbers ................................. 7.5. The Number k in Roth’s Theorem ....................... 7.6. Approximation by Numbers of a Special Type ............. 7.7. Transcendence of Certain Numbers ...................... 7.8. The Number of Solutions to the Inequality (62) and Certain Diophantine Equations .......................... 58. Linear Forms in Algebraic Numbers and Schmidt’s Theorem ..... 8.1. Elementary Estimates .................................. 8.2. Schmidt’s Theorem .................................... 8.3. Minkowski’s Theorem on Linear Forms ................... 8.4. Schmidt’s Subspace Theorem ...........................

27 28 28 30 30 31 31 31 32 33 34 36 39 40 40 41 44 45 45 45 48 50 51 53 53 53 54 55 55 56 57 60 61 61 62 63 65 65 66 67 68

Contents 8.5. Some Facts from the Geometry of Numbers ............... 59. Diophantine Equations with the Norm Form ................... 9.1. Preliminary Remarks .................................. 9.2. Schmidt’s Theorem .................................... §lO. Bounds for Approximations of Algebraic Numbers in Non-archimedean Metrics ................................... 10.1. Mahler’s Theorem ..................................... 10.2. The ThueMahler Equation ............................. 10.3. Further Non-effective Results ........................... Chapter 2. Effective Constructions in Transcendental Number Theory ............................................... Sl. Preliminary Remarks ....................................... 1.1. Irrationality of e ....................................... 1.2. Liouville’s Theorem .................................... 1.3. Hermite’s Method of Proving Linear Independence of a Set of Numbers ........................................... 1.4. Siegel’s Generalization of Hermite’s Argument ............. 1.5. Gel’fond’s Method of Proving That Numbers Are Transcendental ........................................ $2. Hermite’s Method .......................................... ..................................... 2.1. Hermite’s Identity 2.2. Choice of f(x) and End of the Proof That e is Transcendental ........................................ 2.3. The Lindemann and Lindemann-Weierstrass Theorems ..... 2.4. Elimination of the Exponents ........................... 2.5. End of the Proof of the Lindemann-Weierstrass Theorem ... ..................... 2.6. Generalization of Hermite’s Identity .................................. 53. Functional Approximations 3.1. Hermite’s Functional Approximation for e+ ............... 3.2. Continued Fraction for the Gauss Hypergeometric Function .............................. and Pade Approximations ............ 3.3. The HermitePade Functional Approximations 54. Applications of Hermite’s Simultaneous .................................. Functional Approximations 4.1. Estimates of the Transcendence Measure of e ............. 4.2. Transcendence of e” ................................... 4.3. Quantitative Refinement of the Lindemann-Weierstrass Theorem ........................ 4.4. Bounds for the Transcendence Measure of the Logarithm of an Algebraic Number ................................ 4.5. Bounds for the Irrationality Measure of K and Other Numbers .................................... 4.6. Approximations to Algebraic Numbers ...................

3

71 73 73 75 76 76 76 77

78 78 78 79 80 80 82 83 84 85 88 90 92 92 93 93 95 98 99 99 100 103 104 106 110

4

Contents

§5. Bounds for Rational Approximations of the Values .. . of the Gauss Hypergeometric Function and Related Functions 5.1. Continued Fractions and the Values of ez . . . . . . . . . . . . . . . . . 5.2. Irrationality of 7r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Maier’s Results . . .. . .. . . .. . .. . . .. . . . .. . .. . . .. . . . .. . .. . 5.4. Further Applications of Pade Approximation . . . .. .. . . .. . . . 5.5. Refinement of the Integrals. . .. . . .. . . .. . . .. . . .. . . . . . .. . 5.6. Irrationality of the Values of the Zeta-Function and Bounds on the Irrationality Exponent. . . . .. . .. . . .. . . . $6. Generalized Hypergeometric Functions .. . . .. . .. . . .. . .. . . .. . .. . 6.1. Generalized Hermite Identities . . .. . . .. . .. . . .. . . .. . . .. . . . 6.2. Unimprovable Estimates .. . .. . . . .. .. .. . .. . . .. . .. . . .. . .. . . .. .. . . . .. . .. . . .. . . .. . . . . .. . . .. . 6.3. Ivankov’s Construction $7. Generalized Hypergeometric Series with Finite Radius . . .. . . .. . . . .. . .. . . .. . . .. . . .. . . .. . .. . . .. . .. . . of Convergence 7.1. Functional Approximations of the First Kind . . . . . . . . . . . . . . 7.2. Functional Approximations of the Second Kind . . . . . . . . . . . . 58. Remarks . . .. . . . .. . . . .. . .. . . . .. . . .. . .. . . .. . .. . . .. . . .. . .. . . .

136 136 139 143

Chapter

146

3. Hilbert’s

Seventh Problem

............................

Problem .................................. 51. The Euler-Hilbert 1.1. Remarks by Leibniz and Euler .......................... 1.2. Hilbert’s Report ....................................... ........................ 52. Solution of Hilbert’s Seventh Problem 2.1. Statement of the Theorems ............................. 2.2. Gel’fond’s Solution .................................... 2.3. Schneider’s Solution ................................... 2.4. The Real Case ........................................ 2.5. Laurent’s Method ..................................... $3. Transcendence of Numbers Connected with Weierstrass Functions ................................................. 3.1. Preliminary Remarks .................................. 3.2. Schneider’s Theorems .................................. ................. 3.3. Outline of Proof of Schneider’s Theorems $4. General Theorems .......................................... 4.1. Schneider’s General Theorems ........................... 4.2. Consequences of Theorem 3.17 .......................... 4.3. Lang’s Theorem ....................................... 4.4. Schneider’s Work and Later Results on Abelian Functions §5. Bounds for Linear Forms with Two Logarithms ................ 5.1. First Estimates for the Transcendence Measure of ab and lna/lnfl ............................................. 5.2. Refinement of the Inequalities (19) and (20) Using Gel’fond’s Second Method ..............................

112 112 113 115 116 120 121 127 128 131 133

146 146 146 147 147 147 149 150 151 152 152 153 155 157 157 158 159 . . 159 161 161 163

Contents Bounds for Transcendence Measures ..................... Linear Forms with Two Logarithms ...................... Generalizations to Non-archimedean Metrics .............. Applications of Bounds on Linear Forms in Two Logarithms .................................... 36. Generalization of Hilbert’s Seventh Problem to Liouville Numbers ....................................... 6.1. Ricci’s Theorem ....................................... 6.2. Later Results ......................................... $7. Transcendence Measure of Some Other Numbers Connected with the Exponential Function ............................... 7.1. Logarithms of Algebraic Numbers ....................... 7.2. Approximation of Roots of Certain Transcendental Equations .............................. 58. Transcendence Measure of Numbers Connected with Elliptic Functions ...................................... ........................ 8.1. The Case of Algebraic Invariants 8.2. The Case of Algebraic Periods .......................... 8.3. Values of P(Z) at Non-algebraic Points ................... 5.3. 5.4. 5.5. 5.6.

5

164 164 165 165 172 172 173 173 173 175 176 176 177 177

Chapter 4. Multidimensional Generalization of Hilbert’s Seventh Problem ....................................

179

31. Linear Forms in the Logarithms of Algebraic Numbers .......... 1.1. Preliminary Remarks .................................. 1.2. The First Effective Theorems in the General Case ......... 1.3. Baker’s Method ....................................... 1.4. Estimates for the Constant in (8) ........................ 1.5. Methods of Proving Bounds for A, Ac, and Ai ............. 1.6. A Special Form for the Inequality ........................ 1.7. Non-archimedean Metrics ............................... $2. Applications of Bounds on Linear Forms ...................... 2.1. Preliminary Remarks .................................. 2.2. Effectivization of Thue’s Theorem ....................... 2.3. Effective Strengthening of Liouville’s Theorem ............ 2.4. The Thue-Mahler Equation ............................. 2.5. Solutions in Special Sets ................................ 2.6. Catalan’s Equation .................................... 2.7. Some Results Connected with Fermat’s Last Theorem ...... 2.8. Some Other Diophantine Equations ...................... .................................... 2.9. The a&-Conjecture 2.10. The Class Number of Imaginary Quadratic Fields ......... 2.11. Applications in Algebraic Number Theory ................ 2.12. Recursive Sequences ................................... 2.13. Prime Divisors of Successive Natural Numbers ............

179 179 180 182 185 187 188 188 189 189 189 192 193 194 195 196 197 199 199 200 201 203

6

Contents

2.14. Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93. Elliptic Functions . . . .. . . . .. . .. . . .. . .. . . .. . . .. . .. . . .. 3.1. The Theorems of Baker and Coates . . . . . . . . . . . . . . . . . .. . .. . . . .. .. .. . .. . . .. .. . . .. 3.2. Masser’s Theorems .. . . .. . . .. . . .. . .. . . .. . . . .. .. . . .. 3.3. Further Results . . .. . . .. . . .. . . .. . . .. . .. . . .. 3.4. Wiistholz’s Theorems 54. Generalizations of the Theorems in $1 to Liouville Numbers 4.1. Walliser’s Theorems .. . .. . . . .. . .. . . .. . . .. . .. . . .. 4.2. Wiistholz’s Theorems . . .. . . .. . .. . . .. . . .. . . .. . . .. Chapter 5. Values of Analytic Functions That .......................................... Differential Equations

Satisfy

. . . . . .

.. .. .. .. .. .. .. . .. . ..

.. .. .. .. .. .. .. .. ..

203 204 204 204 205 206 207 207 207

Linear

............................................... 51. E-Functions 1.1. Siegel’s Results ........................................ 1.2. Definition of E-Functions and Hypergeometric E-Functions 1.3. Siegel’s General Theorem ............................... ...................... 1.4. Shidlovskii’s Fundamental Theorem ............................... §2. The Siegel-Shidlovskii Method 2.1. A Technique for Proving Linear and Algebraic Independence ......................................... 2.2. Construction of a Complete Set of Linear Forms ........... .............. 2.3. Nonvanishing of the Functional Determinant ................. . ................. 2.4. Concluding Remarks §3. Algebraic Independence of the Values of Hypergeometric ............................................... E-Functions 3.1. The Values of E-Functions That Satisfy First, Second, and ...................... Third Order Differential Equations 3.2. The Values of Solutions of Differential Equations of Arbitrary Order ....................................... ............ $4. The Values of Algebraically Dependent E-Functions 4.1. Theorem on Equality of Transcendence Degree ............ 4.2. Exceptional Points ..................................... 55. Bounds for Linear Forms and Polynomials in the Values of EFunctions ............................................... ...... 5.1. Bounds for Linear Forms in the Values of EFunctions 5.2. Bounds for the Algebraic Independence Measure ........... ...... 56. Bounds for Linear Forms that Depend on Each Coefficient ....................... 6.1. A Modification of Siegel’s Scheme 6.2. Baker’s Theorem and Other Concrete Results ............. 6.3. Results of a General Nature ............................. ............................... ‘$7. G-Functions and Their Values 7.1. G-Functions .......................................... ................................... 7.2. Canceling Factorials 7.3. Arithmetic Type ......................................

209 209 209 . 210 213 214 215 215 218 219 221 222 222 225 229 230 231 234 234 237 241 241 243 244 246 246 250 252

Contents 7.4. Global Relations ...................................... 7.5. Chudnovsky’s Results .................................. Chapter 6. Algebraic Independence of the Values of Analytic Functions That Have an Addition Law ............................ $1. Gel’fond’s Method and Results .............................. 1.1. Gel’fond’s Theorems ................................... 1.2. Bound for the Transcendence Measure ................... 1.3. Gel’fond’s “Algebraic Independence Criterion” and the Plan of Proof of Theorem 6.3 ........................... 1.4. Further Development of Gel’fond’s Method ............... 1.5. Fields of Finite Transcendence Type ..................... , .......... $2. Successive Elimination of Variables ................ ............... 2.1. Small Bounds on the Transcendence Degree 2.2. An Inductive Procedure ................................ $3. Applications of General Elimination Theory ................... 3.1. Definitions and Basic Facts ............................. 3.2. Philippon’s Criterion ................................... 3.3. Direct Estimates for Ideals .............................. 3.4. Effective Hilbert Nullstellensatz ......................... $4. Algebraic Independence of the Values of Elliptic Functions ...... 4.1. Small Bounds for the Transcendence Degree .............. 4.2. Elliptic Analogues of the Lindemann-Weierstrass Theorem 4.3. Elliptic Generalizations of Hilbert’s Seventh Problem ....... $5. Quantitative Results ........................................ 5.1. Bounds on the Algebraic Independence Measure ........... 5.2. Bounds on Ideals, and the Algebraic Independence Measure 5.3. The Approximation Measure ............................ Bibliography Index

7

253 255

259 260 261 262 264 267 268 271 271 272 276 276 278 284 287 290 291 . 295 296 302 302 . 304 306

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Notation

9

Notation

W is the set of natural numbers No = NU{O) Z is the set of integers Q is the set of rational numbers IR is the set of real numbers C is the set of complex numbers A is the set of algebraic numbers ZA is the set of all algebraic integers Zz is the set of all algebraic integers of the field lK , z,) is the set of all rational functions in the variables ~1,. . . , z, K(Zl,... over the field K , z,] is the set of all polynomials in the variables ~1,. . . , z, over K[Zl,... the field IK H(P(z)) = H(P) is the height of the polynomial P(z) E @[zr, . . . , z,], i.e., the maximum absolute value of its coefficients L(P(z)) = L(P) is the length of the polynomial P(z) E C[zr, . . . , zm], i.e., the sum of the absolute values of its coefficients deg,, P is the degree in Zi of the polynomial P deg P is the total degree of the polynomial P t(P) = deg P + In H(P) h(1) is the rank of the homogeneous ideal I C iZ[ze, . . . , z,] deg I is the degree of the homogeneous ideal I C Z[ZO, . . . , zm] H(I) is the height of the homogeneous ideal I C Z[ZO, . . . , z,] t(I) = deg I + In H(I) (I(w)] is the magnitude of the homogeneous ideal I c Z[ZO, . . . , z,] at the point w E C*+l degcY is the degree of the algebraic number cr H(o) is the height of the algebraic number & L(a) is the length of the algebraic number o Norm(a) is the product of all of the conjugates of the algebraic number a m is the maximum absolute value of the conjugates of the algebraic number a IQ],, is the p-adic norm of the algebraic number LLI ]]a]] is the distance from the number a E IR to the nearest integer ]]zr]] = maxr
Introduction

11

Introduction

0.1. Preliminary Remarks. The purpose of this survey is to describe the basic concepts, results, and methods of transcendental number theory, along with its relationships to other branches of mathematics. Definition 0.1. A number Q E Ccis said to be algebraic if there exists a polynomial P(Z) E Z[Z] not identically zero such that P(o) = 0. We shall let A denote the set of all algebraic numbers. A complex number that is not algebraic is said to be transcendental. Clearly, a number C is transcendental if and only if the numbers 1, <, . . . , C” are linearly independent over Q for any n.

In addition to theorems about transcendence of certain numbers, transcendental number theory includes irrationality results and also certain types of quantitative results. Among the examples of quantitative theorems are those that give estimates for approximations of algebraic numbers by rational numbers. The authors do not want to hide their fondness for transcendental number theory, but they do not expect that all readers will at first share this feeling. Hence, we shall start out by trying to explain what we find particularly fascinating in this field. In the first place, in transcendental number theory many of the basic problems have very simple statements, but require deep and powerful methods to solve them. In many casesit took hundreds or even thousands of years to find solutions, and in other cases,despite the very simple formulation of the problem, we still do not have a solution. For example, it is still unknown whether e + 7r is irrational. In the second place, the theory of transcendental numbers has an ancient history. The seminal questions were asked by the mathematicians of antiquity. The methods of transcendental number theory largely come from analysis and algebra. It should also be noted that, conversely, some of the results in this field can be used to help solve problems in other areas of mathematics. 0.2. Irrationality of fi. When the mathematicians of ancient times tried to find a common unit of measure for two intervals, they discovered that such a unit does not always exist. For example, the side and the diagonal of a square do not have a common unit of measure (it seemsthat this was first established by mathematicians of the Pythagorean school). This fact is clearly equivalent to the irrationality of 4. A proof can be found in the tenth book of Euclid’s Elements, so the theorem on the irrationality of fi may be regarded as the earliest theorem (with a proof given) in transcendental number theory. In recognition of the ancient history of this theorem, we shall give several proofs of it.

Introduction

12

(1) Suppose that the side AB and the diagonal BD of a square are commensurable, i.e., there exists an interval which fits exactly m times onto AB and exactly n times onto BD, where m and n are natural numbers, and we may assume that (m,n) = 1. By the Pythagorean theorem, n2 = 2m2. Thus, n is even, and we can write n = 2nr, nr E N. Then m2 = 2n:, and so m is also even, in which case (m, n) 2 2. (2) To prove that the side AB and the diagonal BD are not commensurable, we look at the diagram below.

a

h

In the square ABCD we let E be the point of intersection of the diagonal BD with the circle centered at B of radius AB, and we let F be the point of intersection of the side AD and the tangent to this circle at E. We take G to be symmetrical to E with respect to the line AD. It is easy to see that FGDE is a square whose side DE is equal to BD - AB, and whose diagonal FD is equal to 2AB - BD. If AB and BD were commensurable, then there would exist an interval that goes evenly into both AB and BD, and hence evenly into DE and DF, which are the side and diagonal of the smaller square. This construction can be repeated, using DEFG as the starting square and obtaining a still smaller square whose side and diagonal are both multiples of our original unit interval. Continuing in this way, we obtain an infinite sequence of squares whose sides are multiples of this fixed interval. This is impossible, because the sides of the squares approach zero.

Introduction

13

The above argument is essentially saying that the Euclidean algorithm for finding a common measuring unit for two sides (this is the geometric analogue of the Euclidean algorithm for the greatest common divisor of two integers) will go on indefinitely without finding a solution if it is applied to the side and diagonal of a square. (3) We now translate the above proof into the language of algebra. In the notation in the diagram, we let AB = aibi = 1, so that BD = 4 and DE = azb2 = fi - 1. It is easy to see that for any n E N we have anbn = (42 - 1),-l

= (-l)“(u,A

- ‘u,),

If the intervals AB and BD are commensurable, then fi a, b E N. Since anbn > 0, we have anbn = ’

ah -b&l b

u,,w,

E N.

(1)

= BD = a/b, where

> 1 -b’

which contradicts the fact that anbn -+ 0. These computations are related to the continued fraction expansion of fi. From (1) we obtain %+I = WI + %I,

%I+1

= 2%

+%I,

The recursive relation implies that w,/u, continued fraction approximation to fi: l+2+1

1

211 = 0,

Wl =

‘112= 02 = 1 .

is the (n - 2)-th convergent of the

= [l; 2,2, . . .] = [l; 2] = 1 + p11 + p11 + . . . *

2+“. The fact that this is an infinite continued fraction - i.e., the Euclidean algorithm for the side and diagonal of the square goes on indefinitely - is equivalent to the irrationality of fi. 0.3. The Number 7r. The mathematicians of antiquity were always in search of ways of computing the area of a circle or the volume of a sphere or ellipsoid. It was this that led to the famous problem of squaring the circle and to various rational approximations to X. The first question that arises is whether 1 and K are commensurable. That they are not, i.e., that x is irrational, was proved by J. II. Lambert in 1766. There was a gap in his proof, which was later filled in by Lagrange. The problem of squaring the circle is the following question: Starting with the radius of a circle and using only ruler and compass, is it possible to construct the side of a square that has the same area as the circle? It was not until two thousand years later that this question was answered in the negative - when F. Lindemann in 1882 proved that x is transcendental. We now explain the connection between this result of Lindemann and the squaring of the circle.

14

Introduction

We choose a coordinate system that has the radius of the given circle as its unit of measurement. The problem is to construct a segment of length fi or to prove that such a construction is impossible. Now any construction with ruler and compass is a sequence of elementary steps of one of the following types: (1) construct the point of intersection of two (non-parallel) lines; (2) construct the points of intersection of a line and a circle; (3) construct the points of intersection of two circles; (4) draw the line passing through two given points; (5) draw the circle with given center and radius. In addition, one sometimes has to make an arbitrary choice - say, of a point on some line or circle that has already been constructed. We shall say that a point, line, or circle is algebraic if the coordinates of the point are algebraic numbers, the line can be given by an equation with algebraic coefficients, or the circle has center at a point with algebraic coordinates and radius equal to an algebraic number. By working with the equations of lines and circles, we easily see that when the elementary steps listed above are applied to algebraic points, lines, and circles, the results are also algebraic. Note that whenever one has to make an arbitrary choice of a point in the plane or on an algebraic line, one can always choose such a point to have algebraic coordinates. Thus, if it is possible by ruler and compass to construct a square of the same area as the given circle, then the corners of that square will have algebraic coordinates. This would mean that fi - and hence also x - are algebraic numbers. We conclude that the impossibility of squaring the circle follows from the theorem that 7r is transcendental. 0.4. ‘lhnscendental Numbers. References to the existence of transcendental numbers go back many centuries. Certainly Leibniz believed that, besides rational and irrational numbers (by “irrational” he meant algebraic irrational numbers in modern terminology), there also exist transcendental numbers. Euler conjectured that log, b = In b/ In a is transcendental if a, b E Q and this ratio is irrational (the latter assumption excludes trivial cases,such as log, 4). Euler’s conjecture turned out to be an extremely fruitful one for transcendental number theory. Its proof led to the development of a powerful analytic method, which we shall describe in Chapter 3. Incidentally, the first examples of transcendental numbers were not found until 1844 - almost a century after Leibniz. Liouville’s construction of these examples will be given in Chapter 1. It should be clear by now that our field has an impressive geneology. In its early years, transcendental number theory was a collection of diverse facts sometimes quite elegant, often rather difficult to prove, and in general isolated from the rest of mathematics and even from other parts of number theory. It is only in the twentieth century that transcendental number theory has become an important branch of number theory with its own problems and methods. At the same time its applications have become far more numerous.

Introduction

15

0.5. Approximation of Algebraic Numbers. We have already mentioned that the approximation of algebraic numbers by rational numbers is part of transcendental number theory. This statement might seem odd, and even presumptuous, since there is a separate area of mathematics called “algebraic number theory,” and transcendental number theory is supposed to be the study of numbers that are not algebraic. However, there are some important reasons for including this topic in transcendental number theory: (1) In order to prove that certain numbers are transcendental, one often usestheorems on approximation of algebraic numbers; and to prove results on approximation of algebraic numbers, one frequently makes use of theorems on transcendental numbers, as we shall see in Chapter 4. (2) The construction of many examples of transcendental numbers has been based on theorems on approximation of algebraic numbers. For instance, that is what Liouville did in 1844; and K. Mahler’s beautiful example 0.123456789101112131415.. . (whose decimal expansion consists of the sequenceof natural numbers) is also of this type. We now discuss the role that theorems on approximation of algebraic numbers play in the study of Diophantine equations. We begin with a simple example. Suppose that we want to determine whether or not the equation x3 - 2y3 = m,

mCZ,

(2)

has infinitely many rational integer solutions. Let (X, Y) be such a solution. We may suppose that mXY # 0. If we factor X3 - 2Y3, from (2) we find that

It is well known that any real number has infinitely many approximations X/Y, X,Y E Z, to within Y- 2; however, almost all real numbers (in the sense of Lebesgue measure) do not have infinitely many approximations to within Y-2--6, 6 > 0. From (3) it follows that fi is approximated by X/Y to within O(IY le3). If we show that (3) cannot hold for IY I sufficiently large, then this will imply that there are only finitely many solutions to the Diophantine equation (2). The same argument holds for the entire class of Diophantine equations of the form f(x, Y> = m , (4) where f(x, y) is a form with rational integer coefficients. After obtaining a lower bound for the absolute value of the difference between an algebraic and a rational number, the Norwegian mathematician A. Thue in 1908 proved that any such Diophantine equation (in particular, (2)) has only finitely many solutions, provided that the form is irreducible and has degree greater than two. Sixty years later, W. Schmidt extended Thue’s theorem to Diophantine equations of the form

Introduction

16

7% N

Q1(d q -t ... + a,Cd xm

j=l

a!j) E Kc

A,

i = l)...,

m, j = l)...)

12,

>

= M )

degK=n>m,

MEZ.

We shall discuss the theorems of Thue and Schmidt in more detail in Chapter 1. Thue’s theorem was ineffective: it did not give a method of finding all of the solutions of (4). This deficiency was removed 60 years later by A. Baker using lower bounds for the absolute value of linear forms zilnai 3,*..,

We will describe

Baker’s

%I

work

+.-.+z,lna,, E 4

in Chapter

Xl,...,Xm

E z.

4.

0.6. Transcendence Questions and Other Branches of Number Theory. Transcendental number theory has many connections with other areas of number theory. Not only do the methods of transcendence theory influence those of certain other fields, and vice-versa; but even the statements of the problems are often related. Earlier we explained how the purely geometrical problem of squaring the circle reduced to the transcendence of K. We now give two more examples. One of the achievements of transcendental number theory is connected with a problem that goes back to Gauss - that of determining all quadratic imaginary fields of class number 1 (i.e., in which the algebraic integers have unique factorization). In 1948, A. 0. Gel’fond and Yu. V. Linnik indicated how one could use transcendental number theory to attack this problem. Their idea could be implemented once Baker managed to prove the necessary estimates for linear forms in the logarithms of algebraic numbers. This was done in 1969 by Bundschuh and Hock, who proved that there are no other imaginary quadratic fields of class number 1 besides those found by Gauss. Using similar considerations, Baker and Stark obtained an effective bound for the discriminants of imaginary quadratic fields of class number 2. Much later, using methods unrelated to transcendental number theory, Goldfeld, Gross, and Zagier proved an analogous result for imaginary quadratic fields of arbitrary fixed class number. In 1770, Lagrange proved that every natural number can be written as a sum of at most 4 squares. In the same year E. Waring posed the following problem: Prove that every natural number can be written as a sum of at most 9 perfect cubes, as a sum of at most 19 fourth powers, and so on. In other words, for any n 2 2 prove that there exists a number s such that every natural number can be written as a sum of at most s perfect n-th powers. This form of the problem was solved by Hilbert at the end of the last century. As Waring’s problem was studied further, many additional questions were asked. We shall discuss only one aspect of this area of research in analytic

Introduction

17

number theory, namely, the search for a proof of a formula for the smallest number s such that every natural number can be written as a sum of s n-th powers. This value of s is customarily denoted g(n). It turns out that g(3) = 9 and g(4) = 19. Using earlier results of I. M. Vinogradov, in the late 1930’s L. E. Dickson and S. S. Pillai published exact formulas for g(n). In particular, they showed that g(n) = 2n + [(3/2)n] - 2 provided that ]](3/2)“]] > (3/4)n, where ]]z]] denotes the distance of z to the nearest integer. It is conjectured that the last inequality holds for all n 2 5; in any case, it has been verified for all 5 5 n 5 471600000 (Math. Comp. 55, 1990, p. 85). Thus, analytic methods were used to reduce the problem of an exact formula for g(n) to that of proving a fact of transcendental number theory. In 1957, Mahler proved that for any E > 0 there exists C(E) such that ]](3/2)nii > emEn,

n 2 c(e) .

This result implies the formula for g(n) for n > c(ln $). But unfortunately, Mahler’s inequality is ineffective, so we have no way to compute C(E). Effective inequalities have also been obtained, but thus far they have all been too weak to give the formula for g(n). The best effective inequality is due to A. K. Dubitskas [1990], who proved that ]](3/2)n]] > (0.5769)n. 0.7. The Basic Problems Studied in Transcendental Number Theory. The first type of problem is to prove irrationality, linear independence, transcendence, and algebraic independence of numbers. We now give the definition of the last of these concepts. Definition 0.2. Numbers
, &n) E Z[Zl,. . . , zml 1

not identically zero, such that P(& , . . . , cm) = 0. If no such polynomial exists, then we say that the numbers Cl,. “, <m are algebraically independent. If any of the numbers C;, . . . , cm is algebraic, then obviously the set is algebraically dependent. The second type of problem in this field is to prove quantitative results. Definition 0.3. The iwationality

measure of a number C is the function

cp(C, W = min I5 - El,

p,qEZ,

O
In Chapter 1 we shall discuss in some detail the important case when < is an algebraic number. Notice that the question of solutions to (2) reduced to finding a lower bound for the irrationality measure of fi. Definition 0.4. Let C = (<so,.. . , cm) E UY+‘. The linear independence measure of the numbers <s, . . . , cm is the function

Introduction

18

IfCo,...,cn are linearly independent over Q, then p(C, H) > 0; otherwise, the function vanishes for H > HO. Since qc - p = q(C - t), we could have used S(<, H) with H = ma+, q), instead of cp(C,X). But we gave the above definition of cp(<,X) out of deference to tradition. The transcendence measure of C is a special caseof the linear independence measure. We take CO= 1,
@CC; H; m) = min P(C) I,

P(z) E Z[z],

15 H(P) < H,

degP 5 m .

We obtain the algebraic independence measure(also called the relative transcendence measure) of a set of numbers wi, . . . , w, by taking the components of C in S(C, H) to be the various products WY . ..wp. y, . . . , V, E NJ: @(WI,. . . , w,; H;nl,. PEqa,...,Zsl,

15

HP)

. . ,n,) = min lP(w)l , 5 H,

deg,,P < nk,

k = 1,. . . ,s .

By H(P) we mean the maximum absolute value of the coefficients of P. There are only finitely many polynomials P E Z[zl , . . . , z,] with bounded degreesand bounded H(P) . If C is a transcendental number, obviously its transcendence measure is positive; while if C is algebraic, then this measure is zero for H 2 HO and m > ms. Similarly, the algebraic independence measure of an algebraically independent set of numbers is always positive. We shall later seethat, given any positive decreasing function W(Z), one can find infinitely many numbers 5 E Iw for which there exist infinite sequences X,, = X,,(c) + 00 satisfying the inequality v(C, &a> < 4m

n= 1,2,...

.

A similar fact holds for the other measures defined above. Theorem 1.9 (see 54 of Chapter 1) enables one to obtain upper bounds for these measures. We have WC07. . . , cm; H) < O(lQI + . . . + (G,J)H’-6(m+1) @(C;H; n) 5 0(1+ ICI + . *. + ICI”)H1-6(“+1)

@(WI,... ,ws;H;nl,..

. ,n,) < 0 fr(l

; ;

+ . . . + IWjlnj)H1--6(nl+1)“.(n.+1)

(5) (6) . (7)

j=l

Here 0 = 6 = 1 for real numbers, and 8 = a, S = 0.5 when not all of the & or wj are real. It is natural to compare any lower bounds we obtain for concrete numbers with the above upper bounds.

Introduction

19

Lower bounds for these measures form a significant part of transcendental number theory. They play an especially important role in applications. Of course, the lower bounds one can obtain depend not only on the particular numbers, but also on the methods used to derive them. We now mention an important inequality. Let C E Cc,and let P(z) = a, fi(z

- ctk) = an9 + . . . + a0 E Z[z],

a,#O;

k=l

L = laol + * *. + lanl . = I[-‘--11; -If the numbers or,. . . , on are distinct, then we have P=I$Fn/<-akl

P2 l-nL2-yn

- l)(l-n)‘2

5 ]P(C)] 5 ponder ]P’(C +

--

T(Ql

-

C))I* (8)

This inequality enables us to go back and forth between bounds for the transcendence measure of C and bounds for the difference 15- a], Q E A. 0.8. Different Ways of Giving the Numbers. It is not likely that anyone will discover an algorithm that in finitely many steps decides algebraicity or transcendence of a number that is given in an arbitrary manner. However, in certain casesthe way in which a number is defined enables one to answer this question. For example, in Chapter 1 we show that Liouville’s theorem gives a method to prove transcendence of certain series. Almost all of the theorems of transcendental number theory give properties of the values at algebraic numbers of certain “good” functions (such as ea, 0 # cr E A), or else the properties of “algebraic points” of these functions, i.e., values /3 E Ccfor which f(P) E A (f or example, 0 # p = In cx for which a! = efl E A). By “good” functions we might mean those that have a power series expansion with algebraic coefficients and also satisfy some other conditions. Since the set of all algebraic numbers is an algebraically closed field, it follows that for any nonzero P(Z) E A[ z ] we have P(O) E A if and only if 8 E A. Lindemann’s theorem, which we will describe in Chapter 2, shows that this property does not carry over to a natural generalization of such polynomials, namely, power series with algebraic coefficients. In fact, the function f(z)

= 5

Pk(z)eQkZ $ 0,

Qk EA,

Pk(Z)EA[z],

k=l,...,

m,

k=l

has a Taylor series at z = 0 with algebraic coefficients. However, if (~1,. . . , om are distinct and nonzero, and if none of the polynomials have common roots, then, by Lindemann’s theorem, f(P) is t ranscendental for any nonzero p E A. One should not, however, think that any entire function whose power series in z has algebraic coefficients will take transcendental values at all algebraic z (except for a certain number of exceptions). We give a counterexample. Let {Pi(z)} be the sequenceof all polynomials in Z[z], arranged in increasing order

20

Introduction

of t(P) = degP+H(P) ( an d in increasing order of H(P) when t(Pl) If rk E Q is a sufficiently rapidly decreasing sequence, then

121 =

nk+l

0,

= deg fi

p,(Z)

= t(P2)).

+ nk: + 1 ,

i=l

is a transcendental entire function with all bt E Q. At the same time we have ~(~1 (a) E A and ~(~1 (r) E Q for all s E No, a! E A and T E Q, since any such value is obtained by summing only finitely many terms of the series. For more details on examples of this type, see [Mahler 19761. Early in this century G. Polya [Polya 19151 studied the growth of entire functions that take integer values on N - functions such as 2z and P(z) E Z[z]. Theorem (Polya). Let f(z) be a transcendental entire function, and suppose that f(n) E Z for all n E No. Further supposethat

where a and A are constants. Then a 2 In 2. Polya’s proof used Newton series. 0.9. Methods. In all of the above problems of transcendental number theory, one is looking for lower bounds for nontrivial linear forms in the numbers under consideration with rational or algebraic coefficients. This is clear in the case of irrationality and transcendence measures; but it is also the case for the problem of irrationality of cr, which is equivalent to the inequality

for all

IP - WI > 0

P, Q E z,

IPI +

I4 > 0 7

and the problem of transcendence of c, which is equivalent to the inequality lanCn + . . . + a01 > 0

forall

a0,...,

a,EZ,

laol+~..+la,l>O.

Thus, in all caseswe are dealing with lower bounds for linear forms. In order to give a lower bound for a linear form C, it is often useful to find linear forms over Z that are not proportional to L but have coefficients that are close to those of ,C, and that are sufficiently small in absolute value. We illustrate this with a simple exampie. Let Cl = a< - b,

Ls = c( - d,

a, b, c, d E Z,

ad # bc .

Then 1 5 lad - bcl = IaL2 - cLl[ < la&I + IcLll . If la&I < 0.5 and c # 0, then

Introduction

21

Similar remarks apply to linear forms in several variables (these will be studied in detail later). In what follows we shall often want to construct (small) approximating forms in the numbers under consideration. It is no exaggeration to say that most of this book is devoted to methods for constructing such forms, as will be clear to the reader who examines any of the subsequent chapters. Over the years a large number of techniques of proof and types of result have accumulated in transcendental number theory. Of course, it is not possible to cover all of this, and we had to choose just a small part of the available material. One should not assume that the most important topics have all been included in the book, and the rest is of limited interest. For example, we had to omit such areas as the classification of transcendental numbers; Mahler’s work in the early 1930’s on the values of functions that satisfy certain functional equations, and the subsequent development of his ideas (for a survey of this branch of transcendental number theory, see [Nishioka 19961); and many results involving non-archimedean metrics. On several occasions we had to shorten the length of this volume. We ended up deleting Chapter 7, which was devoted to the important question of estimates for the number of zeros of a function. We ask our colleagues not to judge us too harshly for our omission of some topics from the book.

22

Chapter

1. Approximation

of Algebraic

Chapter 1 of Algebraic

Approximation

Numbers

Numbers

51. Preliminaries 1.1. Parameters for Algebraic Numbers and Polynomials. Let p(z) =cp(zl)...)

z,)

. .. 2

= c

Vl =o

A,, ,...,v,zp

. . .z:

E C[zl, . . . , z,] .

“, =o

Definition 1.1. The degree in zk, the height, and the length of p(z) are defined, respectively, as follows: deg,,cp(z)

=

L(v)

H(p)

nk;

=

2

. a.

v1=0

=

2

mu

I&,...,v,,,I

IAV1,...,v,,,I

;

.

v, =o

In the definition of degree we suppose that at least one of the coefficients A with vk = nk is nonzero, k = 1,. . . , m. For every cx E A there is a unique polynomial P(z) = u,zn +. *. + a0 E Z[z] ) called the minimal polynomial of cx, such that (1) a, > 0, (2) (as,. . . , a,) = 1, (3) P(z) is irreducible, and (4) P(a) = 0. Definition 1.2. By the degree, height, we mean, respectively, dega = degP(z) = n;

and length of an algebraic number (Y

H(a) = HP(z));

L(a) = L(P(z))

.

, cxtn) of the polynomial P(z) are called the conjugates Therootscx=a(‘),... of Q. They are distinct, and they all have the same degree, height, and length. Their product is called the norm of (Y: Norm a = a(l) . . . otn). Obviously, Q c A. If cr = u/b with a, b E Z and (a, b) = 1, then dega = 1, H(a) = max(lul, lbj), and L(a) = Ial + Ibl. 1.2. Statement of the Problem. Let (Y be a real algebraic number. In the simplest case, the central problem of this chapter can be stated as follows: determine how small 6 = 6(ct; ;’ = IQ - ;I can be for p E Z,

q

E N. In particular, one might want to

a) find out how much is possible, i.e., how close rational numbers can get to a; or

$1. Preliminaries b)

find out how much is impossible,

23

i.e., find a lower bound for 6.

Since Q is everywhere dense in JR, it follows that for any 6 E R (in particular, for any real 0 E A) and for any E > 0 there are infinitely many rational numbers p/q such that p - I’ < & . Thus, questions (a) and (b) are trivial unless we impose some additional conditions. But these questions become very nontrivial for irrational a if we bound q from above and refine (a) and (b) as follows: A)

Find a positive the inequality

non-increasing

function

cp(z) = cp(z, a), 2 E N, such that

IQ - iI I 949) B)

has infinitely many solutions (p, q) with p E Z, q E N. Find a positive non-increasing function Q(z) = Q(z:, a), x E N, such that the inequality

IQ- ;I 2 ti,(Q) holds for all p E Z and q E N with p/q # (Y (or at least for all such pairs with q 2 90). Here we would clearly like to find v(x) that decreases as rapidly and $(x) that decreases as slowly as possible.

as possible,

1.3. Approximation of Rational Numbers. It is easy to get complete answers to questions (A) and (B) f or rational cr. Let Q = a/b, a E Z, b E N, a/b # p/q; then = IQ - ;I . (1) Since (a, b) = 1, by assumption, it follows that the equation ax - by = 1 has infinitely many solutions x, y E Z, and so we can take (p(x) = q(x) = l/(bx). These choices are best possible. Definition 1.3. Let 8 E R, and let W(X) > 0 be a function on N that approaches zero as x + co. We say that 0 has a rational approximation of order w(q) if for some c = ~(0, W(X)) the inequality

0 < le - tj < cw(9) holds for infinitely many pairs (p, q) with p E Z, q E N. (Note that one could include the constant c in the function w(x); however, it is more convenient, for example, to speak of an approximation of order qm3 than to speak of an approximation of order 0.0731qm3.)

24

Chapter 1. Approximation

of Algebraic Numbers

Thus, what we just established is that any rational number has a rational approximation of order q-l and does not have a rational approximation of any higher order. 1.4. Continued Fractions. It is well known that every irrational real number 0 can be represented by an infinite continued fraction (rational numbers are actually represented by finite ones)

whereaeEZandui,uz,... and the numbers

E N. The ok are called the partial quotients of 0,

P7l -=uao+2L+...+II 4n IQ

Id

(Pm%) = 1, 4n E w 7

(2)

are called the convergents. If 8 is irrational, then the continued fraction representation is unique. In 1798, Lagrange [1867] proved that for irrational 8 the convergents satisfy the inequality 1 Qn(Qn

+

Qn+l)

1 < tbP” <-, 972 I QnQn+l I

n= 1,2,...

.

(3)

The numbers p, and qn satisfy the recursive relations Pn+1

In particular, inequality

= %zt.lPn

+ Pm-l,

97x+1

= %+14n

(4)

{qn} is an increasing sequence. From (3) and (4) we obtain the 1

1 < 0-P” <-. (an+l+2kI: Qn I %2+19: I An important consequence of (5) is Theorem

+qn-1.

1.1.

(5)

The inequality

has infinitely many solutions for any irrational real number 0. That is, any irrational real number has a rational approximation of order qm2. From (5) it also follows that there exist numbers having a rational approximation of arbitrarily high order. Namely, it is clear from (2) that qn +pends only on a~, . . . , a,, and not on a,+~. But (5) implies that the upper bound for 10- p,/q,l depends on a,+~. Hence, given an arbitrary positive function w(x), if we choose a,+1 to be sufficiently large in comparison with qn, we can obtain

$1. Preliminaries

25

I I 6- E

< w(qn) .

In the theory of continued fractions one proves that if a rational number p/q with p E Z, q E N, satisfies the inequality

then it is a convergent of the continued fraction for 8. This, along with (5), implies that one can determine the order of rational approximation of a given real number if one knows its partial quotients ok or at least their asymptotic behavior. Unfortunately, this information is known for very few numbers. We mention another important consequence of (5). Let k > 2, and suppose that for a given 19there are infinitely many solutions of the inequality

e - F c q-k . I 1 We already know that when q is sufficiently large these solutions are convergents. Let us denote these solutions Pl/Ql, P,/Q,, . . ., where Qt+i > Qt, Qt = qnt. Then from (3) we have

and hence Qt+l > 0.5Qf’

.

(6)

Thus, the denominators of “good” approximations of a real number must grow rather rapidly. 1.5. Quadratic Irrationalities. Lagrange [1867] proved that, if a! is a real algebraic number of degree two, then its partial quotients are periodic from some point on, and, in particular, are bounded. From this result and (5) we have Theorem

that

1.2. Let (Y E A, dega = 2. There exists a constant c = c(a) such

I I a-;

> cq-2.

Of course, the theorem is obvious if Im(a) # 0. Thus, by Theorem 1.1, a real quadratic irrationality has a rational approximation of order qe2; but, by Theorem 1.2, it has no higher order rational approximation. It is sad to have to admit that the above results seem to exhaust the possibilities for using (5) to answer questions (A) and (B) for (Y E A. We have not yet been able to find the full continued fraction expansion for a single

26

Chapter

1. Approximation

of Algebraic

Numbers

algebraic number of degree 2 3. Computations of the first several thousand partial quotients for such numbers as ti and ti support the conjecture that the sequence of partial quotients is unbounded. Thus, to study the order of rational approximation of algebraic numbers of degree 2 3 one needs other methods. The earliest result in that direction was obtained by Liouville (see [1844a], [1844b]). 1.6. Liouville’s Theorem

Theorem

and Liouville

Numbers

1.3. Let a E A, deg Q = n 2 1. If o # p/q, then c(a)

= (1+

~a~)l-“n-‘L(a)-l

.

Proof. When n = 1 the theorem follows from (1). Let n 2 2. If la-p/q1 2 1, the inequality in the theorem holds for any c E (0,l). Suppose that la-p/q1 < 1, in which case Ip/ql < (CX]+ 1. Let P(z) = a& + . . . + au be the minimal polynomial of cr. Then

Hence.

la-;1

>c(a)lq”P(;)lq-?

Since 12> 2, all of the roots of P(z) are irrational, and hence A = qnP(p/q) 0. But since A E Z, we have IAl > 1, and the desired inequality follows.

#

Theorem 1.2 is obviously a special case of Theorem 1.3. A consequence of Theorem 1.3 is that for k > deg a: > 1 the inequality

QI
cannot have infinitely many solutions p E Z, q E N. Liouville used his theorem to construct the first examples of transcendental numbers. Let a,m E N, a 2 2, t E NO,

6’ = 2(-1)La-“!, k=O

Then

2

m+t = x(-l)k@, k=O

Qm+t

=

Jm+t)!

.

31. Preliminaries

9024007

27

o 0 are called Liouville numbers. The number 0 that we just constructed is a Liouville number, as are the sums of the series In nk+l

Ek = fl,

lim -

k+oo

One can also use continued fractions or infinite products to numbers. It is easy to see that the set of Liouville numbers in fact, the set of numbers in (7) is already uncountable. measure of the set of Liouville numbers is zero, as follows theorem of A. Ya. Khinchin ([1964], Theorem 32).

hnk

= 00.

(7)

construct Liouville is uncountable But the Lebesgue from a well-known

Theorem 1.4. Szlppose that f(z) is a positive continzlous finction on (a, co) for some a > 0, and suppose that xf (x) is a non-increasing function. Then for almost all real numbers C (in the sense of Lebesgue measure) the inequality

c-i
many solutions,

provided

(8)

that the integral

J

am f (xPx

(9)

diverges. If the integral (9) converges, then (8) has only finitely for almost all real numbers C.

many solutions

This theorem implies, for example, that almost all real numbers have a rational approximation of order 1/(q2 In q , but only a set of measure zero have an approximation of order l/(q2 In’+ 1 q), 6 > 0. 1.7. Generalization approximation to an number QI, how small q E N) take at z = higher degree and to Theorem

P(Zl,...,

of Liouville’s Theorem. Liouville’s theorem gives a first answer to the question: for a given irrational algebraic a value can the degree one polynomial qz - p (p E Z, a? The theorem can be generalized to polynomials of polynomials in several variables.

1.5. Let (Yl,. . . ,(Y, E A, degcrk = nk, degQ(cri,. Z,)

=

2..-gakl k=Q

,..., k,zf”“z$

. . ,a,)

= n,

E Z[Zl,...,Z,].

k=O

Illllllllllllllllllllllllll lllll lIllllllllllll

28

Chapter 1. Approximation

IfP(Crl,...

of Algebraic Numbers

, wn) # 0, ohm a,)l

Ifym,...,

2

L(Pyn

(10)

k=l

where 6 = 1 ijcrr,...,o, E R, and 6 = 0.5 otherwise. In particular, P(z) E Z[z], a E A, and dega = n, then either P(a) = 0, or else IP(a)l These inequalities

2 L(P)1-6”L(a)-“N,

N = degP(z)

(11)

are used in the proofs of many theorems.

$2. Approximations of Algebraic and Thue’s Equation 2.1. Thue’s

.

if

Equation.

(4) in the introduction.

Numbers

We already encountered the Diophantine equation Let us consider a more general case. Let

f(x, y) = 5

akxkynwk = a, fi(x

- &iv)

(12)

i=l

k=O

be a form over Cc, a,, # 0. Here &, . . . , cn are the roots of the polynomial f(z, 1). We give two very simple but important facts. Lemma 1.1. Let x, y, a, b, c, d E Cc, 6 = ad - bc # 0, X = ax + by, ~1 = cx + dy. Then there exist effective positive constants G = G(a, b, c, d) and g = g(a, b, c, d) such that

G(I4 + IYI) 2 14 + I4 2 &I

+ Ivl> .

In other words, at least one of the forms X or p must be large, i.e., of order

I4 + IYI. Proof. The left inequality is obvious. Furthermore,

lxl + IyI = IdX - bl + IcX - a4 < lad - bcl

-

I4 + PI

Ic’ + Ial lad - bcl + lad - bcl

(IAl + IpI) .

Lemma 1.2. Let f (x, y) be the form (12). If the numbers cl;, . . . , c,, are distinct, then there exist effective constants C = C(f) > 0 and c = c(f) > 0 such that for any x, y E @

cdl4

+ Ivl>“-’

5 Ifb,Y)I

5 Wxl

+ Ivl>“-l

,

(13)

where E = minl
$2.

Approximations

of Algebraic Numbers and Thue’s Equation

Let f(z, y) be a form with phantine equation

integer

f&y)

= an fi(x

coefficients,

29

and let m E Z. The Dio-

- WY) = m

(14)

i=l

is called a Thue equation. When ,we study a Diophantine equation, our maximum goal would be to find all solutions. If it has infinitely many solutions, then by “finding them” we mean that we have a formula that can be used to obtain any solution of the equation (at least in principle - in practice, we might run up against computations that are outside the range of our technology). For example, we can do this for the equation a5 + by = c,

abc # 0 .

a, b, c E Z,

Namely, if d = (a, b) divides c, a = da1 and b = dbl, then the infinite set of integer solutions are given by the formula z = z. + bit,

Y=

YO

- at

,

where t is an arbitrary integer and (20, yo) is a particular solution, which can be found in finitely many steps. Let n = degf 2 2. Let us first take care of the trivial case. If m = 0 and if any of the oi are rational, then equation (14) has infinitely many solutions; whereas if all of the oi are irrational, then it has only the zero solution z = y = 0. In what follows we suppose that m # 0, and the form f(z, y) is irreducible. Suppose that for some m the Thue equation has infinitely many solutions (zk, yr~) E Z2. For each such solution we have, by (13),

c&k(lxkl+ IYklF

I Iml,

&k

=

lFi$n --

Izk

-

ai?/kl

*

But then for at least one value i = ie there is an infinite sequence (zkt, &) Z2 for which 0 <

+kt

-

“io?/kt

I . (bkt

I -I- lykt

I)“-l

5

[ml

E

.

Then the inequality

I,.-!!“-’ < : 20

Q

has infinitely many solutions. But this means that Theorem 1.3 (Liouville’s theorem) gives the exact order of approximation of oio. Now supposethat for at least one value (Yof the oi Liouville’s theorem gives the exact order in q. In other words, suppose that there exists a = a(o) > 0 such that the inequality

30

Chapter

1. Approximation

of Algebraic

Numbers

a-P
(15)

has infinitely many solutions p E Z, q E N. Then from (15) and the right inequality in (13) we obtain

since for Qk sufficiently large we have bkl

5

(1

+

bbk

-

However, f &, Qk) E Z, and so this value is an integer in the range -[o,], .... [cg]. Thus, Thue’s equation has infinitely many solutions for at least one of these numbers (which will be denoted m). 2.2. The Case n = 2. In 51.5 we already saw that when n = 2 Liouville’s theorem gives the exact order in q of the approximation. Hence, for any irreducible polynomial f(z, y) = ax2 + bxy + cy2, a # 0, a, b, c E Z, b2 > 4ac, there exists nonzero m E Z such that the equation ax2 + bxy + cy2 = m has infinitely many solutions (x, y) E Z2. For example, it is well known that if d is any natural number not divisible by the square of a prime, the Diophantine equation x2 - dy2 = 1 (this is usually called Pell’s equation) has infinitely many integer solutions. Moreover, there is a formula that allows one to find any solution. To find this formula it is enough to determine the so-called “fundamental solution” xi, yi . Then the other solutions are given by the formula xn + y&i

= (Xl + yJ2y,

n 6 No ,

where we keep in mind that (*x, *y) is a solution whenever (x, y) is. The fundamental solution (xi, yi) can be found using the continued fraction expansion of &, which is periodic (see 81.5). 2.3. The Case n 1 3. In the last subsection we saw that the upper and lower bounds for lo -p/q1 were “squeezed” by Theorems 1.1 and 1.2, i.e., they differ by only a constant factor. For n 2 3 we do not have this. However, we see from (13) that if we could replace Liouville’s inequality by the inequality

$3. First Version of Thue’s Method

31

then a form f(z, y) having distinct ai would approach infinity as ]z] + ]y] + 00, CC,y E 2’. This would mean that for any m E Z a Thue equation (14) with distinct (~i would have only a finite number of solutions. Actually, one could get away with less: a concrete equation (14) (i.e., one with a fixed form f(s, y)) would have only a finite number of solutions if Liouville’s theorem could be strengthened just for the roots of the polynomial f(z, 1). It was Thue who made the first progress in this direction in 1907-1909. In [Thue 1908a] he examined the equation axr --yr=c,a,b,cEZ,rEN,r>3.

53. Strengthening

Liouville’s Theorem. Thue’s Method

First Version of

3.1. A Way to Bound q8 - p. Let 8 E R,

e u.2;

P,Po E %

fAqo‘oN,

Pc7oZPoQ.

(16)

Then 1 5 ho -pod

= I(eqo -poh

- (04 -p)qol

I Iho -polq + ieq -ho

. (17)

Further let y, 6 E (0, l), 7 + 6 = 1, and w > 0; and let (p(z) and $J(x) be increasing functions that are inverse to one another. If iho

- poi < cw,

cp(Qo) I Q 5 wow ’

(18)

then leqo -polq < y, and, by (17) and (18), we have

(19) Now if we could associate to every fraction p/q an approximating fraction po/qo that satisfies (16) and (18) for some w and cp(x) not depending on p/q, then we would obtain a lower bound for the order of approximation of 8. The main difficulty in realizing this plan is the need to be sure that pqo # poq. This obstacle could be overcome if to each p/q we could associate two approximations ph/qk and pz/q$ satisfying the conditions (18) and the inequality phqi # p{q&. Then at least one of these two approximations could be used for po/qo, and we would obtain (19). Thue ([1908a], [1908b], [1909]) found a method of constructing such approximations for algebraic 8. 3.2. Construction T > 2,

of Rational Approximations

k(rn k=l

for w.

Let r,n E N,

+ l)(r(n - 1) + 1) * * * (r(n - k + 1) + 1) Zn--k , (20) (r - 1)(2r - 1) . . . (kr - 1) Wn(z) = zYJ(l/z)

.

Chapter

32

1. Approximation

of Algebraic

Numbers

Thue proved the following identity in X: xun(xr)

- Wn(XT) = (A - l)z”+‘Rn(x)

)

(21)

where R,(X) is a polynomial of degree rn - 2n with coefficients that “do not grow very fast” as n increases. Let a=

m,

a,bEN;

IQa-PI
P,QEN.

In (21) we set X = aQ/P, and we let t, be the least common denominator of the coefficients of Un(z) (and hence also of W,(z)). After some obvious transformations we obtain wn

-P,

=

Prnantn,

(a&

- f’)2n+1~n

,

(22)

p, = PW,

where the factor P,, is “not too large.” Thus, from a single rational approximation P/Q to o one can produce an infinite sequenceof rational approximations pn/qn. It is clear that how well the rational numbers p,/q, approximate (Y depends on how small a& - P is. Thue proved another identity: wn+l(z)u*(z)

= 2(r + 1)(2r + l)**-(nr + l)(z - 1p+1 ) (r - 1)(2r - 1) . . . (nr - 1) (23) which he used to obtain the important inequality p,qn+l # p,+lq,. 3.3. Thue’s Theorem

- wn(z)un+l(z)

First

Result.

In [1908a] Thue proved

1.6. Let a, b, r E N, r 2 3, m E Z, m # 0. Then the Diophantine

equation ax’ - by’ = m

(24

cannot have an infinite number of solutions. This theorem can be given in a somewhat different form. Theorem

1.6’.

x,y E Z satisfying

There exists a constant C = C(a, b,r,m) (24) must also satisfy the inequality

such that any

I4 + IYI I c. Proof. We sketch the proof of Theorem 1.6. If X, Y E iZ are a solution of (24)) then for CY= m we have

I /I a--

Y X

d-l+Cy'-2-+...+--

Y X

yr-1 XP-1

WI

IXI’ ’

$3. First

Version

of Thue’s

Method

33

from which it follows that the rational number Y/X is an approximation of order r to the algebraic number cr. In the past our rational approximations have always been written with positive denominators; hence, if X < 0 we write Y/X = (-Y)/lXl. Th e relation (25) allows us to determine the function w in $3.1. Let Xi,Yi E Z be a solution of (24). Thue took the rational number Yi/Xi as P/Q, and used (22) to construct an infinite sequence of good approximations to cr. Because p,q,+i # pn+lqn, it follows that for any other solution X2, YZ E Z of (24) with 1x21 > 1x11 one can carry out the plan in $3.1 with p/q = Xs/Yz. But when 1x2) is large compared to 1x11, we get a contradiction between (19) and (25); hence, 1x21 cannot be arbitrarily large, and this means that the equation (24) can have only finitely many integer solutions. Although the title of Thue’s paper referred only to approximation of algebraic numbers, it was in that paper that he proved finiteness of the number of solutions of the Diophantine equation ax’ - by’ = c, T E IV, r 2 3, a, b, c E Z. At the same time the paper essentially contains a strengthening of Liouville’s theorem in the case (Y = m. Thue used his theorem to derive several results. Corollary 1.1. Let h E N, k E 74, k # 0. There exists n E N such that for of distinct integers al, . . . , a,, at least one of the products

any n-tuple

al . ..a.,

(al + k) . . . (a, + k)

has no fewer than h distinct prime divisors. Corollary 1.2. Let k E Z, k # 0, n E N, n > 2. The Diophantine equation xn + (x + k)” = yn has only finitely

Corollary equation

many solutions (x, y) E Z2.

1.3. Let h, k E Z, hk # 0, n E IV, n > 2. The Diophantine

x2 - h2 =

ky"

has only finitely many solutions (x, y) E Z2. Corollary 1.4. Let h, k E Z, hk # 0, m,n equation (x + h)m + (-l)“%? has only finitely

E N, m,n > 2. The Diophantine = kyn

many solutions (x, y) E Z2.

3.4. Effectiveness. Theorem 1.6 leads us to the question of effectiveness. An examination of the proof showsthat the last step in the argument, which led to a contradiction, works only if 1x1 I is greater than some constant that depends on a, b, and m. We also needed the number In 1X21/ In 1x1 I to be sufficiently large. From this point of view we can restate the theorem as follows.

34

Chapter 1. Approximation

of Algebraic Numbers

Theorem 1.6. There exist constants C and c which can be computed for given a, b, and m, such that either all of the solutions (X, Y) E Z2 of (24) satisfy the condition 1x1 5 C, or else there exists a solution (Xl, Yl) E Z2 for which 1x11 > c 7

and in the latter case any other solution (X2, Y2) E Z2 satisfies 1x21

5

lXlC

*

From this version of Thue’s theorem it immediately follows that all integer solutions of (24) satisfy the inequality

PI+ IYI 5 co. However, unlike in the case of c and C, one cannot say that Co can be expressed explicitly in terms of a, b, and m. This constant also depends on Xi, about which we usually know nothing. For this reason we say that CO is a noneffective constant, and so Thue’s theorem is non-effective as well. Although it gives us finiteness of the number of solutions of (24), it does not lead to an algorithm for finding the solutions. In what follows we shall frequently return to the question of effectiveness. 3.5. Effective Analogues of Theorem 1.6. In [1918] Thue published the first effective versions of Theorem 1.6. He proved Theorem

1.7. Let r be an odd prime, a, b, (Y,p, y E N, and

ua’ - bfi’ = y , (4aa~)~-2

,

y2r-2T~2(r-1)-‘(aarb-lp-r)2r-4+2/r

(26) .

(27)

If p, q, k are natural numbers satisfying the inequality

I& - br I 5 k , then q < AkB .

(28)

Here A and B are constants that depend only on a, b, Q, p, y, and r. Thue gave explicit formulas for A and B, but we shall omit them, because they are quite cumbersome. Of course, (28) gives an upper bound for the solutions of the Diophantine equation ax’ - by’ = m (29) for any m. From this it is also easy to show that ]axf - byr] is bounded from below by a function of 1x1+ Iy] that approaches infinity as Ix] + ]y] + co, x,y E z.

$3. First Version of Thue’s Method

35

Thus, (28) gives effective bounds on the absolute value of the solutions of (29), provided that for given a and b we can find (Y, /3, y E N satisfying (26) and (27). Thue gave two examples when such (Y, p, y can be determined. 1. Let t E N. For (26) we take the equality (t + 1) . l3 - t . l3 = 1 . Here a = t + 1, b = t, T = 3, (Y = p = y = 1, and condition (27) holds for t > 37. Hence, for any h E Nc and m E Z the integer solutions of the equation (38 + h)x3 - (37 + h)y3 = m can be effectively bounded by (28) with 2. For (26) we take the equality

Ic = m and suitable

A and B.

1. 37 - 17. 27 = 11 . The condition the equation

(27) holds; hence, for any m E Z all of the integer solutions X7

are effectively

bounded.

- 17y7 = m

In particular,

of (30)

Thue proved that from the inequality

Ip7 - 17q71 < 106,

P,QEN,

it follows that q < 14293. Using Thue’s examples and the inequality (28), it was possible to give an effective refinement of Liouville’s theorem for the numbers qm, t E N, t 2 37, and m. In fact, in the case of (30) the inequality (28) takes the form q < AIrnIB which

implies

the following

= Alp7 - 17q71B ,

bound for the irrationality

Here C > 0 is also an effective constant.

It remains

measure of m:

to note that

7-$<7=degm. One can similarly derive a bound for the irrationality measure of j/m. We note that if we choose t = u3 or t + 1 = v3, we can obtain bounds for the irrationality measures of the numbers qm and qm, for example, $h%, $‘%, m, m, and so on.

36

Chapter 1. Approximation

of Algebraic Numbers

Thue himself did not see any need to derive these bounds. One might try to explain his attitude by noting that it was Diophantine equations that interested him, and approximations of algebraic numbers were for him a secondary concern. However, Thue’s publication in 1909 of a general theorem on rational approximation of algebraic numbers (which we shall examine in detail in 54) would seem to cast doubt on this explanation. In [1977], p. 248, Thue remarks on the non-effectiveness of his theorem (Theorem 1.12 in our numbering), and it is surprising that in his 1918 paper he did not exploit the opportunity to construct effective examples. For 65 years Theorem 1.7 seems to have hypnotized mathematicians (including Siegel; see [Siegel 19701 or [Siegel 1937]), who failed to notice that the possibility of an effective refinement of Liouville’s theorem for certain numbers was contained in [Thue 19181. This was first pointed out in 1983 by Bombieri and Mueller [1983]. In [Siegel 19371, Siegel noticed a connection between the auxiliary polynomials used in [Thue 1908a] and [Thue 19181 and hypergeometric functions, and was then able to obtain a new result. Suppose that n E N, p runs through the prime divisors of n, and n

A, = 4

n~p’I(p-‘) (

.

Pb

1

Ifa,bEZ,cEN,n>3,and

WI-

1+n/2

->

XnC2n-2

)

then the inequality axn -byy

SC

has at most one solution in relatively prime natural Siegel gave several concrete examples: 1. The equation 33x” - 32yn = 1

numbers

with n = 7, 11 or 13, has only one integer solution 2.Leta,b,c~Z,n~N,nz3,and

x = y = 1.

Then there is only one pair of relatively the Diophantine inequality

prime natural

x and y.

numbers

that satisfies

axn - bynj 5 c . 3.6. The First Effective Inequalities of Baker. In 1964 a series of papers by A. Baker ([1964b], [1964c], [1967a]) gave effective theorems on approximation of algebraic numbers that involve roots of rational numbers, along with bounds for the solutions of the corresponding Diophantine equations.

$3. First Version of Thue’s Method For any K. > 2 he obtained

the inequality axT - by’1 > CIxl-

where

37

a, b, r E N, r > 3, and b/a is “close”

)

to 1, namely, 10n - 5

a > (a - l1)~(3r)~~-~,

p=m.

Here c = (9ar2/2)-0,

a=(~--)

(

ln(9ar2/2) ln(81r4/2)

(r+a-b+1+(2p-5)

From this he derived an effective bound for the solutions equation axT - byT = Ck,lXkY1, ck,l E c

> ’ (31) of the Diophantine z

.

k+l
In addition, if u, v, X, Y E Z and if u~+vvii OJ= XqYi+Yti is irrational, then for any p E Z and q E N we have

I I a-~

9

> Qlq-n,

n>2,

where ~0 = 31-‘(2a)-‘H-“(2

+ Ial)l--nc,

H = m~(l4 I4 1x1,IV) ,

and c is given by (31). In particular, cr can be taken to be m. In some cases (32) can be used to find bounds on rational approximations to roots of natural numbers. For example, Baker obtained the inequalities

I

fi _ ; > 10-6q-=‘55; I

m - iif > 10-gq-2.4 9I I

In deriving these bounds an important role was played by relations of the type (26) with small y. In the case of these examples, these relations were 2 . 43 - 53 = 3,

183 - 17. 73 = 1 .

In order to prove his inequalities Baker used the identity A,(x)

- (1 - z)~B,(x)

= x~~+‘C,(X)

where nEN,

v=i,

n>3,

mEN,

O<x
,

(33)

38

Chapter 1. Approximation

of Algebraic Numbers

F(-v+m+l,m+1,2m+2,z)

Cm(x) = An(l) F(-v+m+l,m+1,2m+2,1)

’

t-1 (a + k)(P + k) F(a,P,y,x) =Fxtkno (Y+k)(l+k) . t=o The polynomials

A,(z)

A,(z)Bm+l(z)

and Bm(z)

satisfy

- Am+l(z)&n(z)

the identity

= x2m+1

m k2 - ,,2 v(m!)4 II (2m)!(2m + l)! k=l --P--.

(34) The identities (33) and (34) are analogues of Thue’s formulas (20) and (21)‘. For small x = 1 -b/a, a, b E N, (33) can be used to obtain an infinite sequence of good rational approximations to m that are given by explicit formulas; by (34), the rational approximations corresponding to two successive values of m are different. In the years that followed, a number of authors obtained effective lower bounds for the irrationality measure of cr = m for A E Q and r E N and upper bounds for solutions of the corresponding Diophantine equations. G. Chudnovskii published many such results (see, for example, [1979b] and [1983a]). Below we list some numerical results obtained in [Easton 19861 by D. Easton (based on work of Chudnovskii). The inequality

holds for

In E. M. Nikishin 2/c), then

q3

2.2.10-s

2.795

0

@

1.03. lo-l7

2.405

1O1g76

m

7.81 . lo-lo

2.629

0

m

7.8. 1O-7

2.9099

0

m

1.46. lo-l1

2.313

0

[1982] proved that, if E > 0, m E N, and lnm > 11.4(1+

$3. First Version

of Thue’s

Method

39

Here p and Q are arbitrary natural numbers, and c = c(m,~) > 0 is an effective constant. This inequality is a consequence of a more general theorem of Nikishin on approximation of roots of certain numbers in Q(i). We also mention some results of A. N. Korobov [1990]: 1. Ia - p/q1 > 4-2.5, P, Q 6 N, Q # 1, Q# 4. 2. All solutions (z, y) E Z2 of the Diophantine equation 2x3 - y3 = M,

MEZ,

satisfy the inequality

I4 + IYI I M2 . 3.7. Effective Bounds on Linear Forms in Algebraic Numbers. In [1967a] Baker also obtained lower bounds for the absolute value of linear forms in several roots of natural numbers. Let a,b,mr,... ..,mk3,and ,mk,n E N; hml,. x = a(a - q-“-14-“k(“+l)

> 1.

Thenforanyze,zr,...,zkEjZwehave X0 + X&/b)ml’n

+ *. . + x&%/b)““‘”

> Cx--& ,

where

x = oyjyk I4 ’ 03 -c =

IC= (Iclna-nk(k+l)ln4)ln-‘X,

8-n-‘A-55(n+l)k-(k+l)(n+l)

.

As a corollary he proved that for m E N, m > loll, 1 = m5 - 1, and M E Z, all of the integer solutions of the equation x5 + ly5 + 12z5 + 51x2yz2 - 51xy3z = M satisfy the inequality

14,Ivl, I4 < e500M2. Similar results were obtained by other authors. In particular, effective lower bounds were found for linear forms of the form L = xl(q + alyy

+ . . . + xm(q + a*$”

)

where al,... , a,, q,y E &, K is an imaginary quadratic field, IqI is much larger than (ail,. . . , la,], xi,. . . ,x, is a nontrivial m-tuple of elements of &, and v E Q. For certain values of Y the lower bounds for IL1 made it possible

40

Chapter

1. Approximation

of Algebraic

Numbers

to derive effective upper bounds for the solutions in & analogue of Thue’s equation:

of a multivariable

NormL=f(zi,...,z,). Here f(zi, . . . , 2,) is a fairly low degree polynomial over Zx. E. M. Matveev, who proved several theorems of this type in the nonarchimedean as well as archimedean metrics (see [1979], [198Oa], [1980b], [1981], [1982]) obtained effective bounds on the solutions of certain ThueMahler equations [Matveev 198Oa] Norm L = p”f(zi,

. . . , z,),

f E qxl,...,Gn]~

The method used in the multivariable caseis a generalization of the method described in $3.1. Similar arguments will be used to prove some of the theorems in Chapter 2, so that we will later have the chance to examine this method in more detail. Recently, a series of new effective lower bounds for the absolute value of linear forms of the form

where a,b,Al,... ,A,,B A. K. Dubitskas [1990].

$4. Stronger

E N and x0 ,...,

x,

E Z, have been obtained by

and More General Versions of Liouville’s Theorem and Thue’s Theorem

4.1. The Dirichlet Pigeonhole Principle. In the general case, when o is an arbitrary algebraic number of degree n 2 3, Thue was not able to obtain explicit formulas for polynomials analogous to the ones in (20), so he used a different method. In the last century Dirichlet [1842] showed how to use a very elementary argument to obtain important and rather precise results on Diophantine approximations. This argument, often called the “pigeonhole principle,” is that if n+ 1 objects are placed in n pigeonholes, then at least one of the pigeonholes must contain two or more objects. We shall give several examples of theorems whose proofs use this principle. Theorem 1.8. Let al,. . . , aN E R, XI,. . . , XN E N, and X = maxXj. Then there exists a nontrivial N-tuple x1, . . . , xN E Z (‘nontrivial” means not all zero) such that

54. Stronger Versions of Liouville’s and Thue’s Theorems

Theorem

41

1.9. Let M, IV, X E N, N > M,

Li = 5

ER

aij

&,jXj,

-4 > 5

i = l,...,M.

lai,jl,

j=l

j=l

Then there exist x1,0,. . . , XN,o such that

I-Woo>l I AiX 1 - N/M Theoreml.lO.

) i = l,...,M;

O< lFjyN -- Ixj,Ol 5 x .

Letm,nEN,m
AMEN,

i=l,...,

m.

j=l

Then the system

of equations Q,lXl

+

. . . +

. . . am,121

has a solution

alpxn

=

. . . +

. . . +

am,nxn

=

(x1,0,. . . ,xn,o) E Zn that satisfies

0 < l~~:n Ixj,o I L (AI . . . A,)l/(-)

0 . . . 0

the condition

5 A’d(“-“1

4.2. Thue’s Method a E A,

in the General

Case. Let

dega = n > 3, 6 > 0, N E N,

Using an analogue of Theorem PN(x),

QN(x)

A=II$.~mlAjl. --

7

--

M = [N(S - 1 + n/2)]

.

(35)

1.10, Thue proved that there exist polynomials

Ezkl,

Ez[x, a] ,

RN(x)

such that J'N(x)

d%fk(X),

- ~QN(x) d%QdxC) L(QN),UPN)

= (x -

# 0,

ajNR~(x)

I M + N;

deg,

RN(X)

5 DN ,

where D = D(a, S) is an effective positive constant. Let p E Z, q E N. If Iqa - pl < 1, then, setting PN

=

PN(p/q)q”+N,

qN

= &iv@/q)q”+N

(36) I M ;

(37)

Chapter 1. Approximation of Algebraic Numbers

42

and using (35), (36) and (37), we obtain an infinite sequenceof approximations to (Y. We can also obtain a series of complementary approximations if we use the identities that are obtained from (36) by differentiating several times these are the rational numbers

We already saw that we want to ensure that the condition (16) holds. In the previous case, Thue used (23). He did not have such an identity in the general case, so he developed a different technique, generalizations of which still play an important role in transcendental number theory. Let V&N(X)

=

p;‘(,)Q:‘(x)

-

@‘(x)&$(x),

s,tEN,

s#t.

Vs,t,~(z) is a polynomial with rational integer coefficients and with degree at most K = 2M + 2N - s - t. Thue proved that Vs,i,~(z) is not identically zero and is divisible by a high power (say, the Z-th power) of the minimal polynomial of (Y. Thus, if V~,I,N(Z) is divisible by (qz-p)h, we must have 0 5 h 5 K-nl. This implies that h is not very large, i.e., p/q cannot be a root of VO,~,N(X) of very high multiplicity. It is easy to show that the derivatives of VO,~,N(Z) can be expressed as a linear combination of the polynomials VB,t,~(~), from which we see that there exist a, b E No that are not very large and are such (p/q) # 0. Thus, we can take the pb, qh and pz, qi in $3.1 to be that Va,b,N P$)(p/q)q”+N-“/al

-7 Q$‘(p/q)q”+N-a/a!

P$“(p/q)q”+N-b/b!,

.I

Q;‘(p/q)q”+N-b/b!.

The rest of the argument proceeds as in $3.1. Using the above considerations, Thue [1908] proved the following general theorem. Theorem 1.11. Let f(z, y) E Z[z, y] be an irreducible form Then for any M E Z the equation

f(Xc,Y> = A4

of degreen 2 3. (38)

has only finitely many solutions (x, y) E Z. Remark. The irreducibility condition can be replaced by the weaker requirement that the polynomial f (z, 1) h ave at least three distinct roots. If this condition fails, then the corresponding equation (38) might have infinitely many solutions. Simple examples of this include the equations (z + y)” = 1 and (z2 - 2~~)~ = 1. The latter equation is satisfied by all pairs (xm, ym) E Z2 given by 2, + fiy, = (3 + 2fi)m, m = 0, fl, f2,. . .. Theorem 1.11 has been used to answer several number theoretic questions. Thue [1917] and Mordell [1922] proved, respectively, that the equations

$4. Stronger

Versions

ax2+bx+c=dyn,

of Liouville’s

and Thue’s

a, b, c, d, n E Z,

Theorems

43

n 2 3, a(b2 - 4ac) # 0

and ey2=ax3+bx2+cx+d,

a,b,c,d,eEZ,

ae#O

have only finitely many integer solutions (in the second equation one must assume that the cubic on the right has distinct roots). In particular, the Diophantine equation x2 - y3 = m, m E Z, m # 0, was shown to have only finitely many integer solutions. This equation had been the subject of research for three centuries. Polya [1918] proved that if a,b,c,x E 2 and a(b2 - 4ac) # 0, then the greatest common multiple of the numbers ax2 + bx + c approaches infinity as z + 00. In particular, this is true of the polynomial x2 + 1 (as had been conjectured by Gauss). Thue proved that, if P(x, y), Q(x, y) are forms in Z[z, y], P(x, y) is irreducible, and deg P > deg Q, deg P > 2, then the Diophantine equation P(x,Y> = Q(x,Y) has only finitely many solutions. Here is one more corollary of Theorem 1.11. Suppose that n E N, n 2 3, and the strictly increasing sequence {zk} consists of perfect squares and perfect n-th powers. Then LFa(zk+l - zk) = 00 . We shall give one example with a proof. Let P be the set of natural numbers all of whose prime divisors belong to a certain finite set (~1, . . . ,p,}. We show that the equation x-Y=c,

CE z,

c#O,

(39)

has only finitely many solutions X, Y E P. Suppose that x, y is a solution of (39), with x =pY’ . ..ymy.

yzpp

. ..p$-.

pi,4

E No *

If we set pi = 3ai + ui,

Vj

ai,ui,bj,Wj

=3bj+Wj,

z =p;‘...p”,“,

t =pi’

. ..pk

E No, Ui,wj 5 2, )

then Az3 - Bt3 = c , where A and B are integers taken from a finite set. By Thue’s theorem, each of these equations has only finitely many solutions. Since there are only finitely many equations, this means that the solutions of (39) are bounded. However, the bound on the absolute value of the solutions is non-effective, since Thue’s theorem is non-effective.

44

Chapter 1. Approximation

of Algebraic Numbers

4.3. Thue’s Theorem on Approximation of Algebraic Numbers. In [1908] Thue did not go beyond proving Theorem 1.11, although he already had the tools needed to strengthen Liouville’s theorem. He published his improvement on Liouville theorem in [1909]. Theorem 1.12. Let Q: E A, n = dega 2 3, and E > 0. There exist effective positive constants qo = qo(a,e) and a = a(a,e) such that ijpp1/ql and pz/qz satisfy the inequality

Q_ ?1< q-l-E-n/2 ) I QI

(40)

where then (41) Remark. In Theorem 3 of [Thue 19091,the inequality qz 5 G = G(q) was given rather than (41); however, from the proof it was clear that In qz/ In q1 was effectively bounded. An immediate consequence of Theorem 1.12 is Theorem 1.13. Suppose that the conditions of Theorem 1.11 are fulfilled. There exist constants QO = Qc(o, E) > 0 and ~0 = ~(a, E) > 0 such that 1) the inequality (40) has no solutions with q 1 Qo; and 2)foranypEZ,qENonehas

Q_; >Qq-l--G. I I

(42)

Thue derived Theorem 1.11 as a corollary of his Theorem 1.12. In fact, one can obtain even more from (42). Suppose that f(z, y) is a form in iZ[z, y], deg f = n 2 3, and (~1, . . . , cr, are the roots of the polynomial f(z, l), where cyi # crj for i # j. Then by Lemma 1.2 If(x:,~)l

2 44 + IYI>“-~ l
Ji)ylJ

x,yEZ,

c>o.

We use Theorem 1.13 with E = l/4 to bound the last factor from below. We obtain (here cl(f) > 0, x, y E Z):

If(

> s(f>(l4

+ Ivl>“-’ * 64 + lYlF’2)-(1’4)

= Cl(f)(lXl

+ lyl)(2n-5)‘4 .

(43)

Now suppose that the degree of the polynomial g(z, y) is lessthan (2n - 5)/4. Then it follows from (43) that g(z, y) h as smaller order of magnitude than f(z, y) as 2, y -+ 00, z, y E Z. Hence, all of the integer solutions of the equation m, Y> = g(z, Y>

(44)

$5. Further

Development

of Thue’s

Method

45

must satisfy the inequality

I4 + IYI I MO. This observation was published by Maillet in [1916]. 4.4. The Non-effectiveness of Thue’s Theorems. The condition (41) gives an upper bound on the denominators of rational numbers that satisfy (40). However, because this bound depends on q1, Theorem 1.12 can become effective only when we are able to bound q1 from above. So far no one has found any examples where they were able to determine a solution pl/ql of (40) for which q1 2 qo, or at least give an upper bound for such a q1. To be sure, in 1982 Bombieri proved a theorem similar to Gel’fond’s theorem (which we shall give in s5.2) without the condition q1 2 qo. This led to a series of effective inequalities, which we shall discuss in $5.4. Thue himself commented on the non-effectiveness of his theorems (see [Thue 19771,p. 248). Despite the lack of any effective upper bounds for the solutions themselves, it is fairly simple to find effective upper bounds for the number of solutions to equation (38) and inequality (40). We shall describe such results in 57.8.

55. Further

Development

of Thue’s Method

5.1. Siegel’s Theorem. We have used Thue’s identity (36) to set up some good rational approximations to a number CLNow we shall use the identity in a somewhat different way. We first introduce a new argument y, and subtract yQ(z) from both sides (we also omit the subscript N). We obtain the identity (x - c~)~R(z) - (y - a)&(~) E P(z) - yQ(z) - T(z, y) . Suppose that pl/ql

(45)

and pz/qz are “good” approximations to a. If we also have (46)

then, since T(z, y) E Z[z, y], we find that (47) In view of (45) and (47), lo - pl/qll and lo - pz/qzl could not both be very small, and we would be able to prove Theorem 1.12. Although we do not have (46), we can obtain

for some s that is not very large. In fact, Thue proved that there exist some not very large values of a and b for which Va,b(pl,ql) # 0 (see 54.2); hence, one of the two numbers

46

Chapter 1. Approximation

is nonzero, so that s Theorem 1.12 follows Instead of (45), in (where N is an integer

of Algebraic Numbers

can either be taken equal to a or to b. The proof of once we bound r8 from below. [1921a] Siegel used the following more general identity parameter, as it was for Thue):

(x - 4Nw&

Y) - (Y - a)Vh

Y) = WC&Y>

(48)

I

where

U(z,Y>, Vb, Y> Ea? As before, the main difficulty not very large and for which

Y, 4,

WhY)

E a&Y1 *

was to prove that there exists

s E No that is

(49) where & and
Wbc,Y> =

f:

h(2)!&(y),

TsS,

(50)

i=O

where the polynomials fs (z), . . . , f~ (2) are linearly independent over Q, as are the polynomials go(y), . . . ,g~(y). From (48) and (50) we obtain the relations

where A(z) is the Wronskian determinant of fe(z), . . . , IT(Z), and Al(z) is the cofactor of the element f:“(z). It is clear from (51) that the polynomial A(z) is divisible by the (N - T)-th power of the fundamental polynomial of a; hence, A(z) cannot have a zero of very large order at cr. This, together with the relation

which follows

from (50), ultimately

enables us to obtain

(49).

$5. Further Development of Thue’s Method

47

Generalizations of this technique were used by Gel’fond, Dyson and Roth in papers that we shall discuss later. The use of more parameters than in Thue’s version - namely, the coefficients of W(z, y) - enabled Siegel to strengthen Theorem 1.13 significantly. Here he also investigated the number of approximations by algebraic numbers. Theorem 1.14. Let (Y E A, dega = n 2 2. 1) Given e > 0, there exists cl = C~(CY,E) > 0 such that q E N one has

11 a - f

> c1q-x,

X=

min t4...,n(&+g

for

+e*

any p E Z,

(52)

Here one can replace X and cl by 2&i and c2 = c~((Y) > 0. 2) Let IK be an algebraic number field of finite degree, degK (Y = d > 2, E > 0. Then there exists c3 = C~(CK,l&c) > 0 such that for any primitive element (’ of the field lK one has

Here one can replace p and cs by 24 and cd = C~(CY,K) > 0. 3) Given E > 0, there exists c5 = C~(CY,E) > 0 such that for any [ E A, deg<=u
IQ - tl >

(&+t> p = u,=$::, , ,

~5H(t)-~,

Here one can replace p and c5 by 2ufi

+e.

and c6 = es(a).

The number 2&i becomes less than the exponent 1 + n/2 in Thue’s inequality (42) as soon as n = 12. Note that (52) is a more precise inequality than (42) for n 2 7; the two inequalities are identical for n 5 6. Siegel derived many corollaries from his Theorem 1.14. We shall give some of them. 1. He extended Thue’s Theorem 1.11 to the case when M, the coefficients of f (2, y), and the solutions to equation (38) are algebraic integers. He did the same for equation (44). 2. In [Siegel 19261 he proved that there are only finitely many rational integer solutions of the equation Y2 = P(x),

P(x)

(5 Q4

P(x)

$0 7

provided that P(x) has at least three distinct roots. 3. Let cr E A, degcr = n. There exists a constant c = c(a) such that for any P(z) E Z[z], P(z) $0, degP(z) = h < n one has IP(

2 ~H(P)-~~‘fi+l

.

48

Chapter 1. Approximation of Algebraic Numbers

For small h (namely, for 2h2 5 fi) this inequality is more precise than (11). However, the constant c is non-effective. 4. Suppose that K is an algebraic number field of finite degree, P(z) E Z,[z], P(z) $0, 0 E Zn<, and iV = N(8) is the maximum of the norms of the prime ideals of lK that divide the principal ideal (P(8)). If P(z) has at least two distinct roots, and if H(0) + co, then N(0) + 00. 5. Siegel used Theorem 1.14 in the proof of his famous theorem on finiteness of the number of integer points on algebraic curves of genus g > 0 [Siegel 1929/1930]. He proved that if lK is an algebraic number field of finite degree, and C is an algebraic curve over Z that cannot be parameterized by rational functions, then C can contain only finitely many points whose coordinates are integers of K. In [1931] Mahler proved Theorem 1.14 for (Y= mm, a, b, m E N, using a method that resembled some ideas of Hermite that we shall discussin Chapter 2. Theorem 1.14 and its corollaries are non-effective, since cl, . . . , cs are noneffective constants. 5.2. The Theorems of Dyson and Gel’fond. The following refinement of Thue’s theorem was proved in [Dyson 19471. Theorem 1.15. Let Q E A, degcu = n > 3, E > 0. There exists a constant c = ~(a, 8) > 0 such that for any p E Z and q E N with q > c one has a--p >q- diik 9 I 1

.

Here also c is a non-effective constant. In [1948] Gel’fond obtained this inequality as a corollary of a more general theorem. We shall give Gel’fond’s theorem in a somewhat more precise form (in Gel’fond’s original version si = s2 = S = 1). This refinement can be obtained without any significant changes in Gel’fond’s original proof. Theorem 1.16. Let crl, (~2 E A, llC = Q(CXI,(~a), deg lK = n, 6 = 1 # K is real and 0.5 otherwise, E > 0, 2 5 t$, e2 < n, &t12 = 2n + E, s E N. There exist efectiwe constants LO = Lo(crl, (Ye,E,&, B2,s) > 0 and a = U((YI, CQ,E,&,&, s) > 0 such that, if two algebraic numbers
--Cl/

CL,

e1ss/s1

7

e2ws2

< L;

b2--<2[

7

where de&l

= 31, degb = ~2, degQ(
=

Ll(Cl)

2

Lo,

L2

then L2 5 L;” .

=

L2(<2)

= )

S,

$5. Further

In particular,

Development

of Thue’s

Method

49

for ~21= a2 = ck and 81 = 82, from (53) it foUows that

Ia! - cL(c)-G-E

,

where c = c(o, E) is a non-effective positive constant. In the next theorem (~1, ~2, n, E, 81, and 82 are as in Theorem 1.16. Theorem 1.17. There exist eflective positive constants qo = qo(crl, a~,&, 81,132)and D = D((Y~, (~2,E, fill, 0,) such that if the inequalities

hold for

PI

,pz

E Z and 41, q2 E N, where q1 2 qo, then Q2

I Qf *

Dyson’s inequality is obtained from this by setting a1 = ~22and 81 = 02. In place of Siegel’s polynomial W(z, y) (see (48)) Gel’fond used the polynomial R(x, y) = 2

2

h=O

Ch,kxhyk

=

k=O

c

&,,(x

-

%)“(Y

-

az)”

,

*+*>1

t,

where ch,k E Z (for Dyson ~21 = as). Here 7, A, and B are suitably chosen natural numbers. The polynomial R(x, y) has an expansion all of whose terms have “large total degree” in (z - ai) and (y - ~2). Then R(
with r and s not very large that is nonzero at the point (
can be bounded from below rather well using (10). Thus, (ai -[I) and (CQ-(2) cannot both be very small at the same time. Since the upper bound for qz depends on q1, Theorems 1.16 and 1.17 are non-effective. Two remarks should be made concerning Theorem 1.17. Let (~1,(~2 E A and deg Q(ai, (rz) = n. 1. Let deg (~2 = h. If ~1 has a rational approximation pl/ql for which

I

cY1- pi

I

< qLB1,

4 > 2,

q1>q.,

50

Chapter 1. Approximation

of Algebraic Numbers

then, by Theorem 1.17, for any E > 0 a rational cannot exist for which m-2 I

I


approximation

e 2 =2n+E -7

pz/qz for as

q2>c.

01

Here c is an effective constant, since we know ql. If, in addition, 2n < 01h - which holds, for example, if (~1 E Q(as) - then we will have obtained an effective refinement of Liouville’s theorem for as, and consequently also an effective bound on the solutions of the corresponding Thue equation. But until now no such (~1 have been found. 2. Theorem 1.17 is on the borderline of effectiveness. Suppose that for n 2 3 we were able to replace the condition 9182 = 2n + .e by the condition 01e2 = 2n - 6 for some 6 > 0. Let degcr2 = n, (~2 E IR. We choose ~1 in the field Q(a2) and not in Q. Then, by Theorem 1.1, the inequality 0 < a1 -Es
e1=

2,

has infinitely many solutions. If we choose a solution with q1 > 42, we obtain an upper bound for the values of q2 for which

I I a2-;


-e2=-n+i>-n,

and this bound is effective, because we can find the solution we need with q1 by computing sufficiently many convergents of the continued fraction expansion of ffi. 5.3. Dyson’s Lemma. In [Dyson 19471the inequality (49) was proved using a certain lemma. This lemma of Dyson and its generalizations have played an important role in subsequent research. Lemma 1.3. Let R(z,y) E U&y], R(z,y) $0, deg, R 5 u, deg, R 5 s. Suppose that the (n+l)-tuples XQ, . . . , xn and ~0,. . . , yn each consist of distinct numbers, and let 6, A, to,. . . , t, E Iw with 0 < 6 < 1, X 2 2/f5, 3 < d(u + 1)/2, X[ti+l]
i=O ,...,

0 5 ti 5 s , n.

If 0 I i L n,

0 5 v 5 ti,

0< - p 5 X(ti - v) ,

then X2(1

i=o

+ [ti])(Zti

- [ti]) 5 (2 + n(n + 1)6)(S + l)(U + 1) .

55. Further Development of Thue’s Method

51

We know, of course, that a system of M homogeneous linear equations in M + 1 unknowns has a nontrivial solution. In this case the unknowns are the coefficients of the polynomial R(z, y), so that M + 1 = (s + l)(u + 1) is the number of unknowns. The left side of the last inequality in Dyson’s lemma is equal to twice the number of equations obtained by setting the derivatives equal to zero as in the hypothesis of the lemma. Thus, the “extra element” on the right in the inequality is just the term n(n + 1)s. Because of the 6, this term can be made arbitrarily small. Meanwhile, we have not imposed any conditions on zi and yi (except that the numbers in each (n + 1)-tuple are distinct). Thirty-five years later, in [1982] Bombieri used a similar lemma to obtain an analogue of Gel’fond’s Theorems 1.16 and 1.17, but without the Le and (70. 5.4. Bombieri’s Theorem. Bombieri formulated his results in a general way that included both the archimedean and non-archimedean norms. We shall not do this here; rather, we shall give a narrower formulation, and shall limit ourselves to rational approximations. Theorem 1.18. Let IK be an algebraic number field of degree n 2 3, and let 13, t, and r be positive real numbers satisfying the conditions 0<\/2-nt2<7
constants =

Q(a2)

(54)

c = c(n) > 0 and ~0 = q,(n) =

Pl,PZ

K

E z,

Ql,QZ

> 0 such that

if

E N,

then one of the following three inequalities must hold:

I

Ql

-

a2 I lnq2+

I -; 2 E

>

I

(cH(c%))-

nW-&+r~

(lp1l + q1)-&

(CH(a2))--(2--~,(,-,)

2 ln(c0H(a2)) n(2 - nt2) n < nt2 + 72 - 2 Wal

)

(lp21+ q2)-z% )

< (57)

+ a) +

n(‘-J:

&2)

WCH(Ql))

.

In other words, if neither (55) nor (56) holds, i.e., if both pl/ql and pz/qz are good approximations to ai and cry~,respectively, then it follows that (57) must hold. This means that qz has an effective upper bound in terms of ql. This situation had arisen in earlier papers; however, this was the first time that there was no condition ql 1 qo. That is, if we can find an algebraic number (Y that has at least one sufficiently good rational approximation, not necessarily with large q, then we will have effective estimates (that are better

52

Chapter 1. Approximation

than in Liouville’s

theorem)

of Algebraic Numbers

for approximations

of all primitive

elements of

Qk+ Let 131and 02 denote the powers of (IplI+ ql) and ( Ipa I+ qz). Then e1& = 4(t - T)-~. From the conditions (54) it is clear that t and T can be chosen so that the difference t - 7 is arbitrarily close to m. This is what gives rise to Gel’fond’s condition 8182 = 2n + E. Bombieri gave a series of examples to illustrate the uses of his theorem. Suppose that m, n E N, m, n 2 3, and (~0 is the root that is greater than 1 of the polynomial Xn - mx+-l + 1. It is easy to see that this root is unique and lies between m - 1 and m. Let (~1 = l/a@. Then 1 1 “l-m <m(m-l)“’ I I i.e., for large m and n the number (~1 has a good rational approximation. In what follows (Y,-,and ~1 keep the same meaning, and cz is a primitive element of the field Q(czo). 1. Let n = 200, m > 101731. Then

I I

a - 5 2 (lOm)-

1010H((.y)-1”“H(p/q)-50

2. Let n 2 40. There exist effective constants such that if m 2 mo and H(p/q) > HO, then a

-

;

I

>

mo = ma(n)

H(p/q)-3g.2574

. and HO = Ho(a)

.

I

3. Let m 2 2561 and n > no, where no is an effective constant. There exists an effective constant HO = Ho(a) such that for H(p/q) 2 HO we have

I I CY- ;

> H(p/q)-O.ggggn

.

In [1983] Bombieri and Mueller used similar considerations to study approximations by algebraic numbers of numbers of the form y = fl for 0 E A, n E N. Here is one of their results. Theorem 1.19. Suppose that n 2 3, a, b E N, X = In la - bl/ lnb, (~0 = m, degcro = n, and Q is a primitive element of the field Q((Y~). If X < 1 - 2/n, then for E > 0 and q 2 qo(E) one has

’ Icx- : I’ q-p-E where

113

,=&+6(e)

86. Multidimensional

Variants

of the Thue-Siegel

53

Method

The constant go(E) is efiective. For In b > 216n2 Inn one can clearly choose a so that this theorem is stronger than Liouville’s theorem.

$6. Multidimensional

Variants

of the Thue-Siegel

Method

6.1. Preliminary Remarks. All of the refinements of Liouville’s theorem described so far were obtained by studying two solutions of an inequality of the form

It was natural to expect that by increasing the number of approximations considered, one could obtain still stronger results. As before, the basic difficulty was in the construction of a polynomial in several variables over Z that does not vanish at the point (pl/ql, . . . ,p,/q,), where pl/ql, . . . ,pm/qm are approximations to our algebraic number (Y, but takes a small value there. It took thirty-four years for this program to be realized: Siegel started it in 1921, and in a certain sense Roth completed it in 1955. 6.2. Siegel’s Theorem. In [1921a] Siegel published Theorem 1.20. Let Q E A, dego = n 2 3, k 2 lr&ippl’s) -(here k can be taken equal to e(ln n + l/(2 Inn))).

I I cd

9


has infinitely many solutions pt/qt,

PEZ,

(58) If the inequality QEN,

qt < qt+l, then

lim *nqt+l -=oo. t-m lnqt

(60)

We have already observed that the denominators of “good” rational approximations to a given number are far apart from one another. From (6) it follows that lnqt+i > (L - l)lnqt - ln2, and Liouville’s theorem tells us that k 5 n. The result in (60) gives much stronger information. Proof. We shall give a sketch of the proof. Suppose the contrary, i.e., that the sequence In qt+l/ In qt is bounded from above. Using the Dirichlet pigeonhole principle (Theorem l.lO), one seesthat for any natural number m 2 2 there exists a polynomial

Chapter 1. Approximation

54

P(Xl,...,X,)

= &xv

of Algebraic Numbers

-~)~“F+1,...+n)

E Z[x~~...~xm]

I

(61)

v=l

where the natural numbers ~1,. . . , T,,, are rather large and the polynomials Fl,... , F, have “fairly small” degrees and coefficients. If we substitute suitably chosen approximations p,, /qnl, . . . ,p,, /qnm for xi,. . . ,x,,,, use (59), and make an appropriate choice of parameters, we can show that the right side of (61) is “very small.” Since P(xi, . . . , z,) has integer coefficients, it follows that either 6 = P(p,,/qnl, . . . ,pn,/qn,) = 0, or else 1612 qi$

-**q,-,N,,

Nj = deg,, P .

We do not have to pin our hopes on the nonvanishing of S, since, as in earlier cases, we can use the following fact proved by Siegel: there exist ~1,. . . , sm-r E Ns (of smaller magnitude than the rj) such that

Now let &(x1,. . . , x,) also a polynomial with similar to (61)

Q(xl,.

denote this derivative divided by sr! . . . s,!. This is rational integer coefficients, and it satisfies a relation

. . ,x,)

= g(xu

- a)‘-y-sYGy(xl,.

. .,xm)

7

v=l

but now with p=

Q(P,,/qn,,...,p,mlq,,)

# 0.

In Siegel’s argument an important role was played by the inequality (58) and the negation of condition (60), i.e., the assumption that In qt+l/ lnqt is bounded. If we now compare the upper and lower bounds for ]p], we obtain a contradiction, thereby proving (60). In the same paper, Siegel extended Theorem 1.20 to the case when cx is approximated by elements c of a fixed algebraic number field. In this case qt must be replaced by H(&). 6.3. The Theorems of Schneider and Mahler. In [1936] Schneider proved that Siegel’s Theorem 1.20 remains true for any k > 2. This holds equally well for approximations by elements of a fixed algebraic number field. In the same paper Schneider proved that if k > 1 and all of the denominators q of the solutions of (59) are powers of a fixed natural number, then (60) again must hold. In [1936] Mahler proved that if k > 1, then (60) also holds in the case when all of the prime divisors of the rational approximations belong to a fixed finite set. In the same paper he proved that, if all of the prime divisors of the pq

$7. Roth’s Theorem

55

belong to a fixed finite set, then for any k > 0 the inequality (59) has only finitely many solutions. In [1937] Mahler proved that if qV = q;qF, qL,qc E N, if all prime divisors of all of the qi belong to a fixed finite set, and if lim lnql - 0 -7 v-600 In qV then (60) must hold for any k > 1. Mahler then obtained tiful consequence of this theorem.

the following

beau-

Theorem 1.21. Suppose that P(z) E Q[z], P(z) $0, and P(n) E N for all n E N. Further suppose that g E N, g 2 2, and

P(n) = a,,t(n)gt(n) + -. - +

%,j E No, an,j < 9 *

a,,~,

Then the base-g number e =

0.Q(l)

* . - qoa2,t(2)

* * * a2,o

. . *

is a transcendental number but not a Liouville number. In particular, if g = 10 and P(z) = z, then the decimal C = 0.1234567891011121314151617~~* is a transcendental number but not a Liouville number. In [Korobov 19901 there is a very precise inequality for the irrationality measures of these numbers. Theorem

1.22.

The inequality

)

I I


8-Z

has only finitely many solutions solutions.

if

c > 1;

if c <

1, then it has infinitely many

$7. Roth’s Theorem In [Roth 19551 the author used several approximations to an algebraic number in a new way, not connected with the condition (60). 7.1. Statement

of the Theorem.

Theorem

Let a be an irrational

1.23.

algebraic number, and let k > 2.

Then the inequality c,-!?
(62)

Chapter 1. Approximation

56

We give two equivalent

statements

of Algebraic Numbers of Theorem

1.23.

Theorem 1.24. Let 0: be an irrational algebraic number, and let k > 2. There exists a constant qo = qo(Ly,k) such that for any p E Z, q E N, q 2 qo, one has

a-?

> q-k.

4I

I

Theorem 1.25. Let (Y be an irrational algebraic number, and let k > 2. There exists a constant c = C(CY, k) > 0 such that for any p E Z and q E N one has

The constants in Theorems 1.24 and 1.25 are non-effective. Roth’s theorem leads to a strengthening of the inequality (43), and this, in turn, makes it possible to prove that there are only finitely many integer solutions of (44) in the case when degg(x, y) 5 n - 3. 7.2. The Index of a Polynomial. As in earlier proofs, the proof of Roth’s theorem relies upon an investigation of the properties of an auxiliary polynomial, especially the possible multiplicities of its zeros. To help in this investigation Roth introduced the notion of the index of a polynomial. Definition 1.4. If P(zr, . . . , z,) is a polynomial with complex coefficients, then the index of P at the point (cri, . . . , a,) relative to an m-tuple of natural numbers 91, . . . , g,,, is equal to

r?=Ind(P;ai = min

,...,

a,;gi .+E

(’ 2+-

,...,

. >

gm) =Ind(P;o;g)

,

where the minimum is taken over all ir , . . . , i, pit,...+,

(a,

. . . ,zm>

25=(x

=

il!

=

l . , (&)i' . . * 2,.

E N for which . . . (+J

P(z)

I,=,#

0.

If P E 0, then we take the index to be +a~. If nj = deg,, P, then, by Taylor’s formula, P(Zl,...

,&n)

=

2 *** 2 Pil,...,i,((Y1,...,Qm)(Z1--(yl)il i" =o il =o

. . . (zm

- amp

.

Hence, the index depends upon the initial part of the expansion of P(zr, . . . . . .7 z,) in powers of (zi - al), . . . , (z, - o,). If P(al, . . . , a,) # 0, then we say that the index of P at the point (or,. . . , cr,) is 0. The next fact plays a key role in the proof of Theorem 1.23.

57. Roth’s

57

Theorem

Lemmal.4. LetaEZA,degQ=n,H(a)=H,ql,..., qm,T1 ,..., T,EN, hl,... , h, E Z, (qj, hj) = 1. Suppose that the following inequalities hold:

oak-$ 6r,

10m~0.5m+ 2(1 + 3f5)nJm < 0.5m ;

> 10,

&j-l

>

Tj,

j =2,...,m,

b2 lnqi > 2m + 1 + 2mln((H + l)(]o] + 1)) , j=2,...,m. rjlnqj 2 rl lnql, Then there exists a polynomiaZ Q(zI, . . . , z,) E iZ[zl, . . . , z,] 7-1,. . . , rm in ~1,. -*,-zm, respectively, such that: 1. Ind(Q;cY ,..., cr;ri ,... ,T,) 2 y - (2 + 6c5)nfi. 2. Q(h/ql, . . . , hmlqm) # 0. 3. Foranyil,...,i,ENonehas

(64)

of degreeat most

I&i1,...,i, (a, . . . , a)[ 5 q;(1+36)r1. The proof of the lemma is rather complicated. The polynomial Q is a partial derivative of not very high order of a certain polynomial P having a large index at the point ((Y, . . . , CY).The polynomial P is constructed using the Dirichlet pigeonhole principle. The order of magnitude of the required partial derivative of P is estimated using the following fact, which is known as Roth’s lemma. Roth’s lemma. Let m, ql, . . . , qm, r-1,. . . , rm E N, 0 < S < m-l, lnqi > m(2m + 1)6-l, rj-i/rj > S-l, rjlnqj 1 ~1 lnqi, 2 5 j 5 m. Then any poZynomial P E Z[xl, . . . ,x,] with P $0, H(P) 5 qfT1, and deg,, P 5 rj for 1 5 j 5 m satisfies the inequality ind(P;pi/qi,

. . . ,p,/q,;

v-1,. . . , r,)

5 10”62-m .

In [1995] Evertse used a refinement of the so-called Faltings Product Theorem [Faltings 19911 to give a new proof of Roth’s lemma with 10”~5~-~ replaced by 2m36. 7.3. Outline of the Proof of Roth’s Theorem. Suppose the contrary, i.e., that for some k > 2 the inequality (62) has infinitely many solutions. We choose a natural number m satisfying the conditions m > 4nJm,

2m < k(m - 4nfi)

.

We then choose 6 satisfying the conditions in (63). Next, we choose a solution pl/ql of (62) that satisfies the condition (64), followed by solutions that satisfy the condition PZlQ2~ * . . ,p,/q,

58

Chapter 1. Approximation blnqj

It remains to take ~1 E N satisfying

> 2lnqj-1

E N satisfying

.

the inequality

6rilnqi and rz,...,r,

of Algebraic Numbers

> lOlnq,

,

the inequalities

rl ln ql -
9 ln ql lnqj ’

j=2

,...,m.

These numbers then satisfy all of the requirements in Roth’s lemma. Hence, there exists a polynomial Q( zr , . . . , z,) as in the lemma. If we compare the upper and lower bounds on IQ(pi /ql , . . . , pm/qm) 1, we obtain the desired contradiction. From the above outline of the proof it is clear that the theorem can also be stated as follows. Theorem 1.23’. Let Q E A, degcr = n 1 3, Ic 1 2. There exist effective constants m E N, Q, > 0, and c > 0 such that, if (62) is satisfied by rational numbers pl /ql , . . . ,p,/q, with denominators subject to the conditions

Ql >co,

qj > $-If

j=2,...,m-1,

then the denominator q of any solution p/q of (62) with (p,q) = 1 cannot exceed q&n-l. Here one can take m=

GJ = (2mln((H(o)

K )I 4nk k-2

2

+”

+ l)(l + Ial)) + 2m + 1)C2,

c=-,

2 6

where S is any number that satisfies the conditions in (63). In this formulation we can clearly see the reason why the theorem is noneffective: qm is bounded by an expression in q,,+l, which, in turn, is connected with qm-2, and so on. In the end the bound for qm depends on ql, . . . , q,,,-1. In [Esnault, Viehweg 19841Dyson’s lemma was extended to polynomials in several variables. Let a = (al,. . . ,a,), oj > 0, j = 1,. . . , m. We say that a polynomial P E C[zl, . . . , z,] has a zero of type (a, t) at the point . . , cm) E Cc” if c = (Cl,.

for all vectors i E Nr for which aiir + . . . + am& 5 t. Lemma

1.5.

where
Suppose that & = (&,I, . . . , &,,m) E Cc” for ~1= 1, . . . , M, for p # V, j = 1,. . . , M; tl,. . . , tM 2 0, d = (dl, . . . , dm) E

57. Roth’s

Theorem

59

IV, dl 2 d0 2 ..a > d,,, > 0, a = (al,. . . , a,) E lRm, al,. . . , a, > 0. If there exists a polynomial P E C[zl , . . . , z,], deg,, P = di, having a zero of type (a, tp) at the point <,, for p = 1,. . . , M, then

5

VolMd,

a,&)>

I

fi

I+

(M’

p=l

A!!

- 2) 2

,

dj

i=j+l

j=l

where m

I = I(d, a, t) =

M’ = max(M, 2),

x E I,

Ix

i=l

and Vol(I)

denotes the volume of the solid I.

This lemma was proved using techniques of algebraic geometry. The authors then applied their lemma to prove Roth’s theorem. In [1985] Viola found a different proof of this lemma for m = 2. He was able to remove the condition that &,j # max(c, 9lnn), where c is a constant. If the rational numbers pl/ql, . . . ,p,/q, satisfy the inequality a-p
K>2,

(65)

where Qi+1

>

then 26 nQii-3&+

$7

p = 2m!m2n ,

(66)

142 l-4 + 2) 3ml lnqi

’.

(67)

We now show how this theorem implies Roth’s theorem. Let K = 2 + E, E > 0. We choose m large enough so that the first term on the right in (67) becomes less than 2 + 0.5~. This is possible, since that term approaches 2 as m + co. If the inequality (65) had infinitely many solutions, then one could find m solutions satisfying (66), and q1 could be chosen to be so large that the second term on the right in (67) is less than ~/2. But this would give us a contradiction.

60

Chapter 1. Approximation The notion

of Algebraic Numbers

of the index of a polynomial

was used by Vojta

[1991] in his

proof of Faltings’ famous theorem about the Mordell conjecture. In this connection we also note the papers [Bombieri 19901 and [Fakings 19911. In conclusion, we mention that, after proving Theorem 1.23, Roth was awarded the Fields Medal at the International Congress of Mathematicians in Edinburgh in 1958. 7.4. Approximation of Algebraic Numbers by Algebraic Numbers. We have already encountered this problem, for example, in Theorems 1.14 and 1.16. The simplest (but not the most precise) approach to this problem is to use (10). We shall regard the difference (Y - p, o,/3 E A, as the value of the polynomial zi - zs at the algebraic point (a, p). If Q # /I, then from (10) we obtain

where n1 = dego,

n2 = deg P,

n = deg Q!(w P) I nlm

,

and 6 = 1 if (Y, p E R and l/2 otherwise. In the special case when Q(o) = Q(p), i.e., ni = ns = n, we obtain estimate: Ia - pi 2 2-%(o)-%(/3)-6 .

a good

When CY,,f3 E Q this inequality is not far from the best possible inequality Roth’s method can be used to prove

(1).

Theorem 1.26. Let K be an algebraic number field of finite degree, and let (Y be an algebraic number not in K. Then only finitely many elements C of R satisfy the inequality

This result was stated by Roth, but the first published proof is in [LeVeque 19611. If we do not require that the approximating numbers C belong to a fixed field of finite degree, then we have the following theorem of Schmidt [1970]. Theorem

1.27. Let cx E A, Ima

= 0, v E N, E > 0. Then the inequality

IQ -
by only finitely

many algebraic numbers

(68) C of degree not exceeding

Schmidt’s theorem can be improved only by changing the next theorem from [Wirsing 19611.

E, as follows

from

§7. Roth’s

Theorem

61

Theorem 1.28. Let (Y E A, dega = u + 1 > 3. There exists an effective constant c = c(a) > 0 such that there are infinitely many algebraic numbers < of degree < v that satisfy the inequality

(5x 0.5, l, ifIma=O if Ima # 1

Ia -
0.

In [1970] Schmidt also proved that if the approximating numbers C are algebraic integers, then the exponent on the right in (68) can be replaced by -u-&. 7.5. The Number k in Roth’s Theorem. If we compare Theorems 1.1 and 1.24, we seethat the number Ic = 2 + E, E > 0, cannot be replaced by 2. Thus, in some sense Roth’s theorem gives a best possible result. However, we might hope that the arbitrarily small constant E can be replaced by a function of q that approaches zero. No one has yet been able to do this, but there have been some conditional results, starting with the following theorem from [Cugiani 19591. Theorem 1.29. Suppose that LYE A, dega = n, and the inequality

Ia-Cl
Jln In In H(C) ’

has infinitely many solutions 0 E IK, H(cj) = Hj, Hj 5 Hj+l. lim

j+m

ln Hj+l --00.

Then

_

1nH.j

Cugiani’s theorem was strengthened using the lemma of Esnault and Viehweg (see s7.3). Namely, one can take

lnlnlnH(C) . In In H(C)

This result was proved by Bombieri and van der Poorten [1988]. 7.6. Approximation by Numbers of a Special Type. In $6.3 we gave several theorems of Schneider and Mahler that concerned approximation of algebraic numbers by rational numbers of a special type. One can use Roth’s method to prove these theorems and more general results in an unconditional form. That is, one can dispense with (60) and simply prove finiteness of the number of solutions. For example, we have the following theorem. Theorem 1.30. Let (Y E A, p, E Z, qn E N, (pn, qn) = 1, qn+l > qn, and Qn = 4;4:, where all of the prime divisors of the numbers qx belong to a jixed finite set, and

62

Chapter 1. Approximation

of Algebraic Numbers

lim --w. lnq:, _

wm7 In qn Then for any k > 1 + w the inequality

has only finitely

many solutions.

7.7. Transcendence of Certain Numbers. The theorems of the previous suhsections give us a means to construct new examples of transcendental numbers. Theorem 1.31. If the base-g expansion infinitely many segments of the form

(each segment consists

of 0 is nonperiodic

of n, groups with v, digits in a group), &nF

u

and contains

and if

vunu lim ->o, u+fx A,

=O,

where X, is the number of digits between the decimal point and A,, transcendental. Theorem 1.32. Let Qn be the least common denominator numbers a,,, a,, , . . . , a,,, , and suppose that the series f(z) has positive

= Fa%z--, n=O

mn E

m,+l

NO,

then 0 is

of the rational

> mn ,

radius of convergence R. If lim n-m

m,+l m,

P E z,

=c>l,

lim In’, n+w m,

Q E N,

0 < Ipj < Rql-l”

then f (P/q) is either rational or transcendental. a,,, are positive, then f (P/q) is transcendental.

= 0 7 ,

If p > 0 and the coeficients

Theorem 1.33. Let a, c, d, E Z, a 2 2, c > 2, and Id”/ 5 D”, constant. Then the function f(z) takes transcendental

= 2 a-CYd,zY v=o

values at all nonzero rational

points.

where D is a

$7. Roth’s Theorem Theorem the number

63

1.34. Let 2 = p/q, p E Z, q E N, p # 0, Ipl < q”.5-E, E > 0. Then co c x2” v=o

is transcendental. For additional above theorems,

material on this theme, including see [Schneider 19571.

the proofs of several of the

7.8. The Number of Solutions to the Inequality (62) and Certain Diophantine Equations. Although the general theorems proved by the Thue-SiegelRoth method are non-effective, it is possible to find effective upper bounds for the number of possible solutions to these inequalities and the corresponding Diophantine equations. To derive the simplest estimates it is sufficient to use (6) and the constants go and a given in Theorem 1.12. A stronger result was obtained in [Davenport, Roth 19551. Theorem 1.35. Let Q E A, degcw = n 2 3, 6 E (0,1/3). Then the number of rational numbers p/q, (p, q) = 1, q E N, that satisfy the inequality

I I a - ;

< 0.59-2-6

is less than 26-l In (3 + ln((]a?] + l)(l

+ H(o))‘)

+ exp(70n2F2)

.

Theorem 1.36. Let f(x, y) E Z[x, y] be a form of degree n > 3, and let g(x, y) E Z[x, y], degg 5 n - 3. Then the number of solutions of the equation

f 6%Y) = in rational

965 Y>

integers x and y is less than (4H(f))2”2H(g)3

+ exp(643n2)

.

In the same paper it is shown that, if {Pn/Qn} is the sequence of convergents in the continued fraction expansion of an algebraic number (Y, then In In Qn < c(a)n/

Inn .

It seems that Siegel [1929/1930] was the first to raise the question mating the number of solutions to Thue’s Diophantine equation

f(X,Y) = A4,

of esti-

(68)

with f (x, y) an irreducible form over Z of degree n 2 3, where the bound depends on M but not on the coefficients of the form. The first result of this

64

Chapter 1. Approximation

sort was obtained

only in [Evertse

of Algebraic Numbers

19831. He found the following

bound for

the number N of primitive solutions of this equation (i.e., those for which C&Y>= 1): N 2 715(1+Ct)3 + 6. 72C:(t+l)

,

where t is the number of prime divisors of M. In [1987] Bombieri and Schmidt proved that N 5 c,gt’+t

.

For n sufficiently large, the constant IJ.J can be taken equal to 215. In this paper, solutions (2, y) and (-z, -y) were counted as just one solution, and the equation had the form If(?Y>l = kf * If M = 1, then t = 0, and we have N 5 con. In this connection that the equation xn + a(~ - y)(2a: - y) . . . (na: - y) = 1, has the solutions For certain k there is at most greater than this

we note

aEZ,

(1, 1)) (1,2), . . ., (1, n). < dega it is possible to give an effective bound such that one rational number in lowest terms that has denominator bound and that satisfies the inequality a I

E < q-” 4 I

.

To do this it suffices to use the explicit formula for the constant D in Theorem 1.17 (this was computed in Gel’fond’s paper), along with (17). This was observed by Schinzel, who showed that if IX E A, degcr = n, and k > 3&& then there exists at most one solution to the inequality

where c is an effective constant. The paper [Davenport 19681 should also be mentioned in connection with this question. Several new results in the same direction can be found in Schmidt’s book [1991].

$$3. Linear Forms in Algebraic Numbers and Schmidt’s Theorem

65

$8. Linear Forms in Algebraic Numbers and Schmidt’s Theorem Estimates. In what follows we shall often use vector nota, om are algebraic real numbers, we shall write z = (oi, . . . , (Y,), ,..., crm),2=(zi ,..., z,),

8.1. Elementary

tion.Ifcri,... v=degQ(ai

L(Z) =B.iiJ=Qlzl

+...+a,z,,

Xl,...,%,

E Z)

and IFI = $$yrn I4 * -By Theorem 1.5, there exists an effective constant ~0 = cc,(S) > 0 such that for any vector ?j = (ql, . . . , qm) E Zm either L(q) = 0, or else

IJm>l L coldl-” .

(6%

On the other hand, it follows from Theorem 1.9 that for any & E N there exists a vector Zj = (q1, . . . , qm) E Z”’ satisfying the conditions

IWI I cl& 1 - m ,

0 < l~iym -- I4 I Q 7

i.e.,

IJml 6 CIWrn 7

(70)

where cl = cl(a) is also an effective constant. The inequalities (69) and (70) are incompatible for m > v and large Iv]; hence, for sufficiently large vectors ij satisfying (70) we must have L(q) = 0. In this case (m > V) the numbers cri, . . . , cr, are linearly dependent over Q, and ql,... , qm are coefficients of a linear relation among them. If m = Y, then (69) and (70) only differ by the factors cc and cl, i.e., in this case they have the correct order in ]ij. If m < Y, then the difference between the two estimates is a power of ]q], and it is natural to try to reduce this. The metric results in [Sprindzhuk 19771 show that the following inequality holds for any E > 0, for almost all m in the senseof Lebesgue measure, and for any pointsE=(ai,...,a,)ElR ~=(21,...,Zm)EZrn,Z#0: IL(:)I

> cqpp--E,

c2 = CZ(E,&) .

If we suppose that algebraic numbers behave the same way in the senseof Diophantine approximation as almost all real numbers, then it is natural to try to increase the exponent in (69), perhaps not to the value 1 - m that it has in (70), but at least to 1 - m - E. In the case m = 2, v 2 3, the progress in this direction was examined in @S-S. When m > 2, after the first result of Hasse [1939], it was Schmidt who was able to go much farther. For a detailed treatment of his results, see [Schmidt 19801.

66

Chapter 1. Approximation 8.2. Schmidt’s

Theorem.

of Algebraic Numbers

The theory

developed

by Schmidt

has produced

many different results. Here we shall only give a few of them. Theorem 1.37. Let al,. . . ,a, be algebraic real numbers that are linearly independent over Q, and let E > 0. Then there are only finitely many vectors z= (Xl,... ,x,) # 0 that satisfy the inequality IQlZl

+ *** + (Y,xml

5 Izll--m--E

)

i.e., for 1~1 > 20 one has

This means that, in fact, the inequality (69) holds with exponent 1 - m E. The next theorem concerns simultaneous approximation, i.e., several real numbers are approximated by rational numbers with the same denominator. Theorem 1.38. Let m 2 1; let (~1,. . . ,(Y, be algebraic real numbers that, together with 1, are linearly independent over Q; and let E > 0. Then the system of inequalities -l/m--E

I~jGn+l--jI

<x,+1

has only finitely many solutions xm+l>xo andanyx1,...,x,EZonehas

,

xl,.

j=l,...,m,

. . , xm E Z, x,+1

-l/m--E ma I~jh+l - xjl > x,+1

l<j<m --

E N. That is, for

.

If m = 1, Theorem 1.38 is the same as Roth’s theorem. Schmidt also proved exact inequalities (to within E) in the case when the bound on the linear form depends not on the maximum absolute value of the coefficients xi, as in Theorem 1.37, but rather on the product of the nonzero xi. The next theorem is a strengthening of Theorem 1.37. Theorem 1.39. Let m 2 2; let ~1,. . . ,a, be algebraic real numbers that are linearly independent over Q; and let E > 0. Then there exist only finitely many m-tuples x1,. . . ,x, E Z for which 1x1 . . . xm-lll+Elalxl

+ . . . + QmXml 5 1 .

In applications it is often necessary to have bounds not only for individual linear forms with algebraic coefficients, but also for products of such forms. We now give one such result. When m = 1 it coincides with Roth’s theorem. This theorem also contains Theorem 1.38. Theorem 1.40. Let m 1 1; let ~1,. . . , (Y, be algebraic real numbers that, along with 1, are linearly independent over Q; and let E > 0. Then there are only finitely many natural numbers q such that

$8. Linear Forms in Algebraic Numbers and Schmidt’s Theorem

67

Q1+EllwlI~ **Il%qll < 1 *

(71)

Note that the above theorems of Schmidt are non-effective. 8.3. Miowski’s Theorem on Linear Forms. In order to assessthe strength of Theorems 1.39 and 1.40, we now examine how small it is possible to make the products of linear forms in those theorems. We find useful information on this question in the next theorem, due to Minkowski [1910]. Theorem 1.41. Let m > 2, and let Ll(Z), . . ., L,(Z) be a set of linear with real coeficients and with nonzero determinant A. forms in xl,...,xm Let 71,...,7, be positive numbers satisfying the inequality rr . . . r,,, 2 IAl. Then the system of inequalities

IJw)l has a nontrivial

Llm1

i=l,...,m-1,

< c,

57,

integer solution x1, . . . , x,.

We consider two special cases of this theorem. We keep the notation of $8.2. 1. Suppose that al,. . . , a, are linearly independent over Q, 71,. . . ,7,-l are arbitrary natural numbers, and rm = la,,, I(rr . . .7,+1)-l. We set Li@) = xi,

i = l,...,m-

1;

L,(3)

= a121 + f.. + (Y,x,

.

Since rr . ..r, = Iornl = IAl > 0, it follows by Minkowski’s theorem that there exists a nontrivial m-tuple ql, . . . , qm E Z such that lmq1+***

+ wd7ml

l&l 5 Gi,

< Tm,

i=l,...,m-1.

We now find that

q: *-L1ILnm

< I%l,

d = m&l, hiI) L Ti .

This inequality has infinitely many solutions, since, by choosing 71,. . . , r,-r sufficiently large, we can obtain arbitrarily small T,,,, and L, (ij) # 0 because of the linear independence of (~1, . . . , cr,. Thus, in Theorem 1.39 if we replace the function (21 . . . x,-I)~, which is unbounded on Z?F1, by a certain sufficiently small positive constant, then we obtain an inequality that has infinitely many solutions. 2. Suppose that Ic = m - 1, ~k+r E N, and 71,. . . ,rk are arbitrary real numbers in the interval (0,l) for which 71 . . . rk = l/?-k+1 . We set Lo

= Xi - (Yixk+l)

i = l,...,k;

Lk+l(Z)

=

zk+l

.

Then ~1 . n. rm = 1 = A, and so, by Theorem 1.41, there exist q, ql, . . . , qk E Z, not all zero, such that Id

5

‘-k+l,

IlWqll I laiq - Qil < Tip

i = l,...,k.

68

Chapter 1. Approximation of Algebraic Numbers

We may suppose that q E N, since when q < 0 we can change the signsof q and qk, and q = 0 is impossible, since then all of the qi would vanish. Note Ql,..., that qi # 0 when ri 5 ]CQ].Because rk+i can be chosen arbitrarily large, and hence 71,. . . , rk can be chosen arbitrarily small, it follows that the inequality

9ll~l9ll

*. . Ilwfll < 1

(72)

has infinitely many solutions q E N. This means that the E in Theorem 1.40 cannot be replaced by zero. Also related to these questions is the following interesting theorem of Peck [1961]. Theorem 1.42. Let Do,. . . ,/& be linearly independent elements of a real algebraic number field of degreen + 1. There exists a constant c > 0 such that the system of inequalities

Iq& - q&I

< cq-‘ln(lnq)-ll(*-l),

i=l,...,n-1,

I9Pn - 9nPol < c9+

(73) (74)

has infinitely many solutions q E N, 91,. . . , q,, E Z. Let us denote cy( = pi/Do, i = 1,. . . , n. Then the numbers 1, (~1,. . . , o, are linearly independent over Q, and (73) and (74) imply a stronger inequality than (72): 9ll~l9ll . . . IIh9ll < cln-l 9 . We emphasize that this holds only in the case when 1, al, . . . , on form a basis of a real algebraic number field of degree n + 1. In this case one has a stronger result than Theorem 1.38 (see [Cassels 19571):

ma I~jx,+l - xjl 2 ~gxi$,

l<j
ci)=co(a)>O.

Of course, this does not contradict Peck’s theorem, since one only has

l’=i”=i, -- Id%- 9iPol= IQPn- 9nPolN q-‘ln . 8.4. Schmidt’s Subspace Theorem. Theorems 1.39 and 1.40 (and their corollaries, Theorems 1.37 and 1.38) can be derived from the following general theorem about subspaces. Theorem

1.43. Suppose that the forms Li(5) = ai,

+ . . . + aj,mx,

= Zii . Z,

i = l,...,m,

have real algebraic coeficients and are linearly independent. Let 6 > 0. There exist a finite number of proper rational subspacesof R” such that every point 3 E Zm that satisfies ILl(Z) . . . L&7)1 < l2l-6

38. Linear

Forms in Algebraic

Numbers

and Schmidt’s

Theorem

69

lies in one of them. By a rational subspaceof IV we mean a subspace that can be defined by linear equations with rational coefficients. Example. Let m = 2, Li(Z) = ox - y, Lo = x. Then for Q E A and for any 6 > 0 Theorem 1.43 implies that all of the points Z = (x, y) = (p, q) E Z2 for which lax - yllxl < piq-6 lie on a finite set of lines y = kx, k E Q. Each such line can contain only finitely many points satisfying the above inequality. In fact, if (~0, qo) and (tpo, tqo), t E Z, are two such points, then ltqocr:- tp0l < m=(ltq0l,

ltPol)-61G701-1,

ItI 2+6 < lQol-1-61Qo~ -pal-l

,

i.e., ItI is bounded from above. We conclude that there are only finitely many solutions, and this is Roth’s theorem. The inequality I4 L m=(lRe4,

IIm 4)

enables us to extend Theorem 1.43 to forms having complex algebraic coefficients. Under the conditions of Theorem 1.40 we set L&C) = xi - CQxm+l,

i=l,...,m;

Ln+l(q

=

GrL+1

*

Let q be a natural number that satisfies (71), and let the vector ?j = be defined by choosing the unique pi E Z such that (Pl,... ,Pmd E r+l i = l,...,m.

IIWIII = lw - Pil,

We have Ipi/ < q, and hence Fiji < q.* If we set 6 = s/2 and use (71) with q sufficiently large, we obtain IL1

(3

* * * Lrzfl

@)I

<

Q+

<

IIr6

According to Theorem 1.43, there exists a finite set of linear forms in xm+i with integer coefficients such that every solution q of (71) gives Xl,..., a vector ?j at which at least one of the forms in this set vanishes. (Obviously, the forms MI (E), . . . , M,(Z) are such a set of forms, where MI(T) = 0, . . ., M,(T) = 0 are the equations that determine the subspaces in Theorem 1.43.) Let y denote the minimum absolute value of the values at the point E = ((Yl, . . . ,Qm, 1) of the forms of this set. Since the subspacesin Theorem * Here and in the sequel the notation A < B means that JAI 5 cl?, where c is some constant. In the present case this constant depends only on the coefficients of the linear forms Li.

70

Chapter 1. Approximation

of Algebraic Numbers

1.43 are proper subspaces, we have Mj(Z) $0, and since cri, . . . , cr,, 1 are linearly independent, we have y > 0. Let M(Z) = alzl + . . . + um+l~m+l be a form in the set that vanishes at ij. Then the inequalities laiq - piI < 1 and IM(E)l 2 y imply that rltlli

I

IM@)llql

=

IaM@)

-

M(V)1

=

la1(QIq

-

Pl)

+ * *. + Gn(snq

-

Pm)1

e (all+.**+laml.

Hence, I?jI is bounded from above, and so Theorem 1.40 is a consequence of Theorem 1.43. Theorem 1.39 also reduces to Theorem 1.43. We note that Theorems 1.39 and 1.40 are equivalent; this follows from the so-called “transfer theorem.” Theorem 1.44. Let Ll(Z), . . . , L,(Z) be linearly independent linear forms inxl,...,x, with real algebraic coeficients, and let cl,. . . , c, E IR with Cl + * * * +c,=o.

(75)

Then for any 6 > 0 there exists a finite set of proper rational subspacesof II%“’ such that every z E 23” for which ILi(Z)l 5 j51C’-6,

i = l,...,m,

(76)

lies in one of them. Theorem 1.43 is a consequence of Theorem 1.44. In fact, any point ?EE Z” for which Ll(Z).. *Ln(3 # 0 must, according to (69), satisfy the inequalities

pql-”

< ILi(Z)l

i=l,...,m,

<< IZI,

where v 2 vi = degQ(ai). (Of course, the other points lie in a finite set of rational subspacesof B”, namely, the subspacesgiven by L,(Z) = 0.) Hence, the solution set for the inequality in Theorem 1.43 is contained in the union of solution sets of the finitely many systems of inequalities lzl”: 5 ILi(Z)l 5 121”:1,

i = l,...,m,

(77)

satisfying the conditions 0 < f$ - 12:< 6/2m,

ci + . . f + CA 5 -6 .

In fact, each of the m identical intervals (Al~l’-“,Bl?~l) that contain the possible absolute values of Li(S) can be divided into subintervals of the form (I~Ipkv’, I~lPk+l,i), where i = 1,. . . , m is the index of the form and 0 < Pk+l,j Pk,i < 6/2m. We can then consider all possible sets of intervals for which the index i occurs exactly once in each set.

$8. Linear Forms in Algebraic Numbers and Schmidt’s Theorem We set ci = cy - (cy + . . . + ck)/m. Then (75) holds, and every solution the system (77) is also a solution of the system of inequalities ILi(T)l

5 lZy-6/2m,

By Theorem 1.44, all of the solutions of proper rational subspaces of I+?.

71 of

i=l,...,m. of the last system

lie in a finite number

8.5. Some Facts from the Geometry of Numbers. In the proof of Theorem 1.44 an important role was played by the properties of the so-called successive minima. This concept is defined for a rather large class of solids (star-shaped solids); however, we shall consider only the special case that we shall need later. Let Li(Z) = i& . Z, i = 1,. . . , m, be linearly independent linear forms in x1,. . . ,x,. We fix cl,. . . , cm E lR satisfying (75). Let 17 = n(Q; cl,. . . , c,) denote the solid defined by the inequalities IL$i?)I = I& . ~1 5 Qci ,

i=l,...,

m,

&,~:EIW”,

Q>l.

(78)

We let XI7 denote the solid obtained by stretching 17 (contracting 17 if X < 1) by a factor of X along all rays from the origin. It is obvious that if X is sufficiently large, then Xl7 will contain all of the standard basis vectors of I%“; and if X is sufficiently small, then X17 will not contain any nonzero integer = Xk(Q) = Xk d enote the smallest value of XI, point. Let &(Q; cl,. . .,c,) such that &fl contains k linearly independent integer points. The numbers X, are called the successive minima of the solid 17. Obviously, Xl,..., 0 <

k(Q) i X2(Q) L ... I L(Q).

Example 1. Let 17 be the square with Xl = x2 = l/a.

vertices

at (&a, &a), a > 0. Then

Example 2. Let II be the parallelepiped with vertices where 0 < c < b < a. Then Xi = l/a, X2 = l/b, Xs = l/c. If the vectors $1,. . . ,&, determined by the relations

at (*a, fb, ztc),

form a basis of IS”, then the vectors

& .q =

1, 0, {

6;, . . . ,zL

if i =j, ifi#j,

are also a basis of I+?, called the dual basis of $1,. . ., $-mThe vectors ~1,. . . ,& used to construct the parallelepiped II (see (78)) obviously form a basis of lRm. Let z;, . . . , Zk be the dual basis. Then the dual parallelepiped 17’ is defined by the inequalities 1~; . ~1 5 Q-ci,

i = l,...,m.

The successive minima of l7* will be denoted X;(Q). Theorem 1.44 can be deduced from the following “theorem last minimum”.

(79) on the next to

Chapter

72 Theorem

1.45.

1. Approximation

of Algebraic

Numbers

Let 17 be the solid given by the inequalities (78), let ‘iii,, . . .

. ,;iis, E Z” be linearly independent vectors, and let i& E Xk17. Let 6 > 0, ind let I be the set of indices i for which ci 2 -612. If An-l(Q) and Q is suficiently

< Q-6

030)

large, then q . a:, = I),

iEI.

This theorem implies that for any sequence of numbers Q that satisfy (80) and increase without bound, the vector Ek = m;(Q) takes only finitely many values. In fact, it follows from the theorem that there exists a vector x E Z”, x # 0, with relatively prime coordinates, that satisfies the equations Et .E = 0, i E I. We fix one such vector, and denote it Es. Since the vectors pi; do not depend on Q, neither does the vector &,. The following inequality relating the successive minima of dual parallelepipeds is due to Mahler:

~;(Q)hn+l-k(Q)

z+ 1.

It follows from this and from (80) that X;(Q) >> Q’, i.e., we always have X;(Q) > 1 for Q sufficiently large. But then among the integer vectors in II* (i.e., satisfying (79)) there cannot be two that are linearly independent; in other words, all such vectors are proportional. One can prove that the denominators of the components of the rational vectors %?L(Q) are bounded by a number that does not depend on Q. If E is their least common denominator, then the integer vectors K and m;(Q) are parallel, since they both satisfy (79). This implies that the set of possible values for the vector W;(Q) is finite. Remark. Roth’s theorem is a special case of Theorem 1.45. For sufficiently large q the inequality (62) implies that for large Q the parallelepiped 17(Q) defined by the inequalities

satisfies Xl(Q) 5 Q-’

with

6 = 1 - 2/n.

In this case Theorem 1.45 tells us that the vectors ~?r = (q, p) and 7i; = (1, (Y) are proportional. This contradiction proves Theorem 1.23. The proof of Theorem 1.45 is a generalization of the proof of Roth’s Theorem 1.23. In particular, one has to define a generalization of the notion of the index of a polynomial, and one has to generalize Roth’s Lemma 1.4. The derivation of Theorem 1.44 from Theorem 1.45 uses techniques from the geometry of numbers; results of Davenport and Mahler play a fundamental role here. We shall not give the details, which can be found in Schmidt’s book [1980].

59. Diophantine

Equations with the Norm Form

73

In [1994] Faltings and Wiistholz use the Faltings Product Theorem [Faltings 19911to give a different proof of Schmidt’s Subspace Theorem. Note that the theorems in this section are non-effective.

59. Diophantine

Equations

with the Norm Form

9.1. Preliminary Remarks. Let IK be an algebraic number field of degree n let K(l) = K, ll@), . . . , K(“) be the conjugate fields, let or,. . . , (Y, E Zx, lei c$) be the conjugate of oi in K(j), and let &) = (c$) , . . . , a$). The Diophantine equation

(81)

n

a1Cd~l+--~+a~)z,

= n( j=l

>

=M

is called the norm form equation. We have already encountered this type of equation. When m = 2 and n 2 3 it is Thue’s equation (38), and some examples with m > 2 were given in $3.7. When n = 1 the equation (81) takes the form alzl+---+a

al ,...,

mz m--M,

a,,MEZ,

21,...,ZmEZ.

The condition d] M, where d = g.c.d.(ai , . . . , a,), is necessary and sufficient for solvability of this equation. If this condition holds, then the equation has infinitely many solutions. In $2.2 we looked at Pell’s equation, where m = n = 2. It has the form Norm(z + &y)

= z2 - dy2 = M ,

and for M = 1 it has infinitely many solutions. Now suppose that m = n > 2 and (~1,. . . , cr,,, are linearly independent. We shall show that for every form F(Z) of the type in (81) there exists at least one M for which (81) has infinitely many integer solutions. We first suppose that at least one of the fields K(j) (say, lK = K(l)) is a real field. By Theorem 1.9, for any X E N there exists a vector z E Ztn), Z # 0, for which lh(ql

I

cx

In

-

,

IZjl

5x7

j’l,...,n,

(82)

where C = C(a) is a constant. It is obvious that for some constant Cc we have j = l,...,n. ILj(z)l I COX, (83) Using (8I), (82), (83), and the linear independence of (~1,. . . , (Y,, we seethat if 5! is a solution of (82), then the integer F(Z) satisfies the inequality

74

Chapter

1. Approximation

of Algebraic

Numbers

0 < pq)I 5 ccp .

(84)

From (82) it is clear that max Ixjl + +oo as X --+ +oo, so that there are infinitely many integer vectors that satisfy the inequality (84). Meanwhile, F(Z) for these vectors can take at most 2CCt-1 values; hence, at least one of these values occurs infinitely many times. Now suppose that all of the K(j) are complex fields, and let K(l) and lK(‘) be complex conjugate fields. From Theorem 1.9 it follows that the inequalities 0 < lLl(~)l

= lL2(?~)15 IReLi(f)l

0 < IL,@)L,(Z)l

5 cfx2--n,

+ IImLi(Z)I ZE z!?,

5 c~X~-~/

,

m=lqI LX,

(85)

have solutions for any X E N. If we use (83) to estimate the remaining n - 2 forms, we can again prove that the values F(Z) are bounded on an infinite sequence of integer vectors 5. Note that this argument does not go through if n. = 2 and IK is complex. In that case the inequality (85) implies that the absolute value of F(Z) is bounded from above for an infinite set of natural numbers X. However, it is not possible to prove that the vector F takes infinitely many values - it may happen that the same vector z corresponds to all of the X. Of course, this is in agreement with the observation that the graph of the equation ax2 +bxy+cy2

= 1,

b2 < 4ac ,

lies in a bounded region of the xy-plane, and so for any ax2 + bxy + cy2 = M,

M the equation

a, b, c E Z ,

has only finitely many integer solutions. We now consider the equation (81) from a different point of view. Let M be the module generated by linearly independent algebraic integers (~1,. . . , a, E K, deglK = n,* i.e., M = {xlal

+-..+~,a,

xl,...,x,

E Z}.

An element p E Zn
+. . . + xmam) = M ,

then we also have * An algebraic number (Y E K is said to be an integer of lK if it is the root of a polynomial anzn + . . . + as E Z[z] with a, = 1. The set of integers of lK is denoted Zx. If both (Y and o-l are in Zx, then we call (Y an algebraic unit.

59. Diophantine

Equations

Norm(yiai

with the Norm Form

75

+. . . + gmom) = M

for y1a1 + . . . +ymam =7@1a1 +..-+x,a,),

Yl,**.,

YmEZ.

from a single solution to (81) one can obtain an infinite set of solutions. Of course, it is easy to find a number M for which there is a solution - simply substitute any integer vector in F(Z). Thus,

9.2. Schmidt’s Theorem. In order to have a better understanding of the conditions in the theorem that will be stated below, we give an example. Let lK = Q(fi, fi), deg lK = 4. Consider the equation F(x, y, z) = Norm(zfi

+ y&

+ z&)

= M .

(86)

If we set one of the variables equal to zero, we obtain the equations ((2x)2 - 6~~)~ = 4M;

4(x2 - 3~~)~ = M;

9(y2 - 2~~)~ = M . (87)

From our earlier discussion of Pell’s equation it is clear that for each of the equations in (87) there exist infinitely many M for which the equation (and hence also the equation (86)) has an infinite number of solutions. For example, we can set, respectively, M=M;,

M=4M;,

M = 9M&

Ml,M2,M3

E z.

In [1972, 19801Schmidt proved a general theorem on equations of the type (81). Let n = degQ(cri, . . . , aLI,), 2 5 m < n, where or,. . . , cr, are linearly independent algebraic integers. The set M = {ZICQ +...+x,a,

xl,...,x,

E Z}

is a module in the field Q(ai, . . . , am), but not a full module. Two modules MI and M2 are said to be similar if MI = aM2 for some cr E A. A non-full module is said to be non-degenerate if it does not have a submodule that is similar to a full module in a smaller algebraic number field that is not Q or an imaginary quadratic field. In the above example it is obvious that the module generated by a, a, and & is degenerate. Theorem 1.46. Let K be an algebraic number field of degreen, let m < n, and letal,...,c-w, be linearly independent elements of ZK. Suppose that the module {xlcrl +...+x,a, xl,...,xm E Z} is non-degenerate. Then the equation (81) has only finitely many solutions. Schmidt proved this theorem using his subspace theorem (Theorem 1.43) and a rather complicated argument. To prove the theorem in the case m = 2, when (81) reduces to Thue’s equation (38), it was sufficient to obtain a better bound than in Liouville’s theorem for the smallest (in absolute value) linear

76

Chapter

1. Approximation

of Algebraic

Numbers

factor of f(x, y). But when m > 2 the situation is different. For a given vector

Z there could be several small linear factors on the left in (81). Thus, Theorem 1.37 does not give enough to prove Theorem 1.46 - one needs information about how small the product of several factors can be. Theorem 1.46 is non-effective. In Chapter 4 we shall discuss some possibilities for making it effective in special cases. In [1990] Schmidt gave an upper bound for the number of subspaces in Theorem 1.43. He later applied this result to estimate the number of solutions of the equation (81). This bound was refined in [Evertse 19951. In [1990a, 19921Schlickewei generalized the main result of [Schmidt 19901to an arbitrary absolute value and also to algebraic integer solutions xi. All of these results have been applied to the study of various Diophantine problems.

$10. Bounds for Approximations in Non-archimedean

of Algebraic Metrics

Numbers

10.1. Mahler’s Theorem. In [1933] Mahler generalized the inequality (50) to the case of a p-adic metric. Theorem 1.47. Suppose that ~1,. . . , P,,, are distinct fixed prime numbers, P(z) E Z[z], degP(z) = n 2 3, (’ E R, & E QPj, j = 1,. . . , m, P(C) = P(<,) = a. * = P(&) = 0, E > 0, k > 0. Then the inequality min(l, I< - dql) . fi

min(L IP - QCjlpj> 5

%4 + d-2*-E

j=l

has only finitely many solutions p E Z, q E N, (p, q) = 1. Theorem 1.48. Suppose that F(x, y) E Z[x, y] is a binary form, n. = deg F(x, y), the polynomial F(1, z) has at least three distinct roots, ~1,. . . , pm are distinct fixed prime numbers, E > 0, k > 0, and Q(x, y) denotes the largest divisor of the number F(x, y), x, y E Z, (x, y) = 1, having the form &” . . . p$, Vl,.. . , v,,, E No. Then the inequality ‘;$;;

,

5 k(lpl + Iql)n-2fi-E

has only finitely many solutions (p, q) E Z2, (p, q) = 1. In particular, the largest prime divisor of the number F(p, q) approaches infinity as Ipl + IqI + 00. 10.2. The Thue-Mahler of the results in 510.1.

Equation. The following theorem is a consequence

$10. Non-archimedean

Approximations

of Algebraic Numbers

77

Theorem 1.49. Suppose that F(z, y) E Z[z, y] is a binary form such that the polynomial F(l, z) has at least three distinct roots, ~1,. . . ,pm are distinct jixed prime numbers, it4 E Z, M # 0. Then the Thue-Mahler Diophantine equation F(z, y) = M&I;’ . . . p$’ has only finitely

many solutions

2, y E Z, (x, y) = 1, 21,. . . , z, E No.

We note that Theorem 1.49 implies that there are only finitely many solutions of the Thue equation (14) in rational numbers (x/z, y/z), x,y E Z, z E N, if all of the prime divisors of z belong to a fixed finite set of primes. Theorems 1.47, 1.48, and 1.49 are non-effective. However, as in the case of the theorems of Thue, Siegel, Dyson, Gel’fond, and Roth, the number of solutions of the corresponding inequalities can be effectively bounded. 10.3. Further Non-effective Results. The refinements of Siegel’s theorem usually are also translated into the language of non-archimedean valuations. One question, in particular, that has been examined is the power of a prime ideal that can divide the values of F(z, y). In addition, non-archimedean analogues have been obtained for Schmidt’s theorem. All of these results are non-effective. In Chapter 4 we shall discuss effective bounds that are related to the theorems on linear forms in the logarithms of algebraic numbers. By way of an example we give a corollary of the p-adic version of Gel’fond’s Theorem 1.16. Theorem 1.50. Let IY, [I,. . . , &,, be elements of an algebraic number field IK, wherecl,...,& are multiplicatively independent, let p be a prime ideal of IK, and let E > 0. Then there are only finitely many m-tuples x1, . . . , xm E Z that satisfy the inequality ordp(a where CJJ= max(lsll,. Chapter 4 contains effective as well.

- C,“l ...<m)

> EX,

. . ,Ix~~). results

that are stronger

than Theorem

1.50 and are

78

Chapter 2. Effective Constructions

Effective

in Transcendental

Number Theory

Chapter 2 Constructions in Transcendental Number Theory

Many results in transcendental number theory are proved using the following principle. Suppose that we want to prove that a number CYis irrational or transcendental. To do this, we would like to find a way to construct a sequence of sufficiently good Diophantine approximations to a - perhaps approximations by rational numbers (see ‘$1.4 below), or else a sequence of simultaneous approximations (see §2), or possibly a sequence of polynomials with integer coefficients that take small values at (Y (see 54.2). If (Y were rational or algebraic, there would be some polynomial P(z) E Z[z], P f 0, having CYas a root. If our sequence of approximations is “dense” enough, then we can use it to eliminate a in the equation P(o) = 0, thereby obtaining a contradiction. In several cases the desired rational approximations can be obtained from the convergents of the well-known continued fraction expansion of cx (see $5.1) or from the partial sums of series that converge to (Y (see $1.1). Sometimes one has explicit expressions for the approximations, perhaps in the form of integrals (see 82). On the other hand, frequently it is possible only to prove that the required approximations exist. These ideas also apply when proving quantitative results - estimates for measures of irrationality, transcendence, and so on. If the approximations can be constructed explicitly, then the results usually turn out to be more precise. This is to be expected, since explicit formulas enable one to use specific properties of the numbers. The present chapter is devoted to explicit methods for constructing Diophantine approximations and the results that can be proved using these approximations.

$1. Preliminary Most of this that enable us transcendental. the ideas in the 1.1. Irrationality

Remarks

section ($51.3-1.5) is concerned with some general results to prove that certain numbers are linearly independent or First, however, we give some simple examples that illustrate introduction to this chapter. of e. The irrationality

of

(1) was essentially proved by Euler. To be sure, he did not state this result; however, in 1737 he found the continued fraction expansion

$1. Preliminary e = [2,1,2,1,1,4,1,1,6,1,.

Remarks

79

. .] = [2,1,11],>i

.

To prove irrationality, it would have sufficed to use the fact that this continued fraction is infinite. This observation was made not by Euler, but rather by Lambert [1766], who also used continued fractions to prove the irrationality of rr. We shall discuss this in more detail in 54.1. The proof of the irrationality of e that is currently the best known - using the partial sums of the series (1) as approximations - was apparently first published in Fourier’s Cours d’analyse [1815]. This proof is based on the fact that for every natural number n where

0 < n!e - A, = 0(1/n),

A,=n!&Z.

(2) k=O

’

If e were a rational number, then we would have an infinite sequence of positive integers with limit 0. This is obviously impossible. 1.2. Liouville’s

Theorem.

Liouville [1840] used (2), the formula e-l

= x(-l)“: TZ>O

and the resulting relation n!e-l - C, = 0(1/n),

where

G=n!pLL

(3)

to prove that e is not a quadratic irrationality. Namely, suppose that we had an equality of the form ae2 + be + c = 0 , (4) where a, b, c are integers not all zero. From (2) and (3) we find that n!(ae + b + ce-‘) - (aA, + bn! + cCn) = 0(1/n),

n>l.

From (4) it follows that the integer d,, = aA, + bn! + CC, = 0(1/n)

.

(5)

This means that d, = 0 for n > no. To obtain a contradiction, it remains to show that there exist arbitrarily large n with d, # 0. Using (2) and (3), we obtain: dn+2 - (n + l)(&+l + A) = 2a, where a # 0, since e is irrational. This implies that for every n at least one of the numbers d,, d,+l, d,,+2 is nonzero. At the heart of the above proof is the construction of the rational approximations A,/n! and G/n! for e and e-l with the same denominator n! (“simultaneous approximation”). The transcendence of e was first proved in

80

Chapter

2. Effective

1873

by Hermite,

Constructions

in Transcendental

Number

Theory

who found a method of constructing simultaneous ratio-

nal approximations to e, e2,. . . , em for any natural number m. What he did was essentially to prove the Q-linear independence of 1, e, e2, . . . , em; we shall describe this proof in $2. 1.3. Hermite’s Method of Proving Linear Independence of a Set of Numbers. In 51.2 we saw how simultaneous approximation was used to prove that

1, e, e-l are linearly independent. We now consider a more general situation. Suppose

that

cl,. . . , <m E IL!,

L(T)

=

a0

+

al<1

+

. . . +

am&;

ao,*..,

U,EZ.

(f-3)

Further suppose that be, bi, . . . , b, E Z satisfy the conditions IUlEl + . . . +~m%al
(7)

uobo + . . . +~mbm#O,

(8)

where ck

=

bock

-

k = l,...,m.

bk,

(9)

From (6) and (9) we have boL(<) = 2

ukbk + 2

k=O

(10)

ukek .

k=l

The right side of (10) is nonzero, since, by (7) and (8), it is equal to the sum of a nonzero integer and a term of absolute value less than 1. Hence, L(c) # 0. Thus, to prove that 1,
Generalization

of Hermite’s

Argument.

Let 7 = (cc,. . . , &) E

, L(T)

=

urJ20

+

u1z1

+.

. . +

um3Jm,

UiEZ.

Suppose that the forms Li = Lo

= 2 j=O

bijzj,

i = 1,. . . ) 772,

bij E Z )

(11)

51. Preliminary

Remarks

81

along with L(Z) are linearly independent; and suppose that

(IJWT>I + . . . + I-L(~>I>

. HA”-’

5 c,

(12)

where

and c is a small positive number depending only on m and &. To be definite, we may take c = (2. m!)-l rnaxs
O
(14

j=l

where A is the determinant of the coefficient matrix of the system of linear forms L(Z), Lj(Z), and Aij is the (ij)-minor of this matrix. Since A E Z, A # 0, and hence IAl 2 1, from this identity we find that 2~ = (4-l

o~iFm l&l -5 IL(<)1 . Am + (IL&)1

+ ..a + ILm(~)l) . HA”-1

5 IL(<)1. A” + c , i.e., IL(c)1 2 c-A-”

.

(15)

This inequality implies that, in particular, L(c) # 0. Thus, if for any L(C) one can construct a system of linear forms (11) that are linearly independent with L and satisfy (12)-(13), then we can conclude that the numbers co,. “, Cn are linearly independent over Q. At the same time we can use (15) to obtain a lower bound for IL(c)1 depending on H = H(L). Note that if we take (0 = 1 in the special case Lk(Z) = brJZk- b/&Q,

k= l,...,m,

A = (-l)“b,m-l(aobo

+ . . . + urn&) ,

then we obtain do0 = (-l)“%,m,

Aoj = (-1)“-‘ajbF-‘,

l<jlm,

so that the identity (14) with i = 0 and with <j substituted for xj turns into the relation (10). The above argument is due to Siegel [1929], who put it in a more useful form as follows. Suppose that EN is a sequence of positive real numbers that approach zero, and AN is an increasing sequence of numbers that approach 00. Further suppose that for all suficiently large natural numbers N one can construct a system of linear form.9

82

Chapter

2. Effective

Li = L@)

Constructions

= 2bijZj,

in Transcendental i=O,l,...,

Number

m,

Theory

bij E Z ,

j=O

such that the following conditions hold: 1) the forms Lo(Z), Ll(E), . . . , L*(Z) are linearly independent, i.e., they form a complete system of linear forms; 2) mai,j lbijl 5 AN, lLi(<)l 5 EN * Akm, 0 5 i 5 m. Then the numbers 6,. . . , I,,, are linearly independent over Q. In fact, if the numbers (b, . . . , cm were linearly dependent, then there would exist a linear form

max ]oj] = H,

H>l,

O<j<m --

for which L(c) = 0. Among the forms Lo,. . . , L, satisfying 1) and 2) we can choose m of them, say LI, . . . , L,, which along with L make up a complete system of linearly independent forms. Conditions (12)-(13) then hold with A = AN, since (ILI(~)I

+...+

ILm(~)I).HA~-’

5 eN’mH

+

0

as N + +oo. It now follows that L(c) # 0, contradicting the choice of L; hence, the numbers CO,. . . , cm must have been linearly independent. Lower bounds for the absolute value of linear forms at the point 3 can be proved in a similar manner. One is not usually able to actually construct the systems of linear forms L,(Z). Frequently one can only prove that the required sequencesexist. This was the case in Siegel’s original paper [1929] on the values of E-functions. Siegel’s method, along with some generalizations and refinements, will be presented in Chapter 5. However, in this chapter we shall consider caseswhen an explicit construction can in fact be carried out. 1.5. Gel’fond’s Method of Proving That Numbers Are Transcendental. The ideas in this subsection were first used by A. 0. Gel’fond in 1929 to prove that en is transcendental (for details, see 54.2). Suppose that C E c and P(z) = a0P + ald+-l

+ . . . + a, E Z[z]

If C is an algebraic number of degree n, then under the assumption that P(C) # 0 one can obtain a lower bound for IF’(C)] in terms of m and H = H(P) = mmg<m lajl* Let Q(z) = bozn + blzn--l + . . . + b, E Z[z]

92. Hermite’s

Method

a3

be an irreducible polynomial having C as a root. Suppose that the integer coefficients of Q have no common factor greater than 1, let
where y = be(l+)<))n-‘. 0 ne can prove more exact and more general inequalities (for example, Theorem 1.9), but this is enough for our present purposes. The proof of (16) is based on the observation that the number R = br . fi P(&) j=l

(17)

is a nonzero integer. If we use the inequality IRI 2 1 and bound from above all of the numbers ]P(&)] for j > 2, we obtain (16). We have the following fact: Suppose that C E Cc and a(iV), T(N) are sequences of positive numbers that increase without bound. If there exists a sequence of polynomials PN(x) E Z[x] such that deg Pjv + ln H(PN)

5 U(N),

0<

lpN(<)I

< exp(-dN)

. T(N))

,

then C is a transcendental number. In fact, if C were an algebraic number of degree n, then, applying the inequality (16) to the polynomials PN and using the inequalities m < U(N) and In H 5 U(N), we would obtain a contradiction. A modified version of the above fact enables one to prove lower bounds for lP(fJ ] depending on deg P and H(P) when C is transcendental. This was first done in 1932 by Koksma and Popken in connection with estimates for the transcendence measure of e” (see Theorem 2.7 below) and by Mahler in connection with estimates for the transcendence measure of x and the logarithms of rational numbers (Theorem 2.9). In Chapters 3 and 6 we describe indirect constructions of sequences of polynomials PN (that are actually much more general) that lead to a proof of transcendence and algebraic independence of certain classes of numbers.

$2. Hermite’s In [1873], Hermite Theorem

Method

proved:

2.1. e is a transcendental number.

Corollary 2.1. If r is a nonzero rational number, then e’ is transcendental. In the more than 120 years since Hermite’s paper was published, several quite different proofs of Theorem 2.1 and its generalizations have been devised.

84

Chapter 2. Effective Constructions

in Transcendental

Number Theory

However, this section will be devoted to Hermite’s method. His method gives the following statement, which is equivalent to Theorem 2.1: for any m 2 1 the numbers 1, e, e2,. . . , em are linearly independent over Q. One uses the .’ technique described in $1.3, in which the construction of the simultaneous. rational approximations to powers of e is based on the so-called Hermite. identity. 2.1. Hex-mite’s

For any polynomial f(z)

Identity.

E @[XI one has the iden-

tity

J

oi e-‘f(t)dt

= F(0) - F(z)emz ,

(18) r ‘I

where F(z) = 2 f’“‘(z), s=o

M = degfb)

,

(1%

as can easily be proved using integration by parts. If f(z) has integer coefficients, then the polynomial F(z) will clearly also have integer coefficients. i Let n be a natural number (that will later be chosento be sufficiently large), and suppose that for some natural number m the polynomial f(z) E Z[z] has the property that fci)(0) = fi)(l) Thenforanyk=0,1,2

,...,

= . . . = fti)(m)

= 0,

O
(20)

mwehave

F(k) = 5 p’(k) i=o

= 5 p’(k) i=n

= n!bk )

where bk is an integer. From (18) we find that (21) If we now suppose that f(z)

has the property that (22)

where c = c(m) is a positive constant depending only on m, then from (21) and (22) we obtain the relations hoe” - bk = O(F/n!),

k=O,l,...,

m,

(23)

which show that the rational numbers hi/be, . . . , b,/bs, all with the same denominator, are good rational approximations to e, e2, . . . , em. Now suppose that e is algebraic. Then for some natural number m and integers a,, . . . , aa one has

$2. Hermite’s amem

+

. . -

Method

85

+ ale + a0 = 0,

# 0.

aOam

(24)

If We set & = ek, k = l,.. . ,m, we seethat the left side of (24) is of the form (6). By (23), the numbers Ek = bee” - bk satisfy (7) for all n sufficiently large. In order to show that (24) leads to a contradiction using the technique in $1.3, it suffices to prove that for n > no (for some no) there exists a polynomial f(x) E a z I sat-isfying the conditions (20) and (22) and the inequality (8), i.e., a,F(m)

+ -.. + aoF

# 0.

(25)

2.2. Choice of f(z) and End of the Proof That e is Transcendental. At present we know many different ways to construct polynomials that satisfy (20), (22), and (25). Here we shall give several versions of the construction, where we preserve the authors’ original ideas but occasionally modify some of the details. It is natural to begin with Hermite’s original construction. 1) ([Hermite 18731). For any integer Y, 0 5 v _<m, the polynomial

fv(x)

= (x(x - 1). . * (z - m))“+i/(a:

- V)

obviously satisfies the conditions (20) and (22); hence, it suffices to show that (25) holds for at least one value of V. If (25) fails to hold for all Y, 0 < v 5 m, then we must have = 0, (26) WIFv(~>ll O
O
= (x(x - 1). f. (z - m>>n . zV,

(27)

If they satisfy (26), where the polynomials F,(x) are obtained from them using (19), then there must exist real numbers c,, 0 5 v 5 m, not all zero, such that m.

Ccvfi(k) = 0, v=o

05 k < m .

(28)

We set 4(z) = X(X - 1). . . (z - m), h(z) = cc, + ~12 + . . . + C,X* $0, and f(x) = ti(~)~h(x). Then F(x) = 2 c&(x) v=o satisfies the relation

F(z) - F’(z) = $“(s)h(z)

,

(2%

86

Chapter 2. Effective Constructions

in Transcendental

Number Theory

and from (28) it follows that F(s) = f(z)g(z), where T 2 1 and the polynomial g(z) is not divisible by 4(z). Comparing degrees in the last equality, we see that r 5 n; but then (29) is impossible. 3) ([Greminger 19301). We choose the polynomials fV(z) in the same way as in 2), and we use the same notation. If (26) holds, then, as shown above, F(z) vanishes at the points O,l, . . . , m (see (28)). The function e+F(z) also vanishes at these points, and, in addition, it approaches zero as z + +co. This implies that the derivative of eeZF(z), i.e., the function -e-“4”(z)h(z) (see (29))) must vanish at m + 1 points <s,&, . ‘0, <m satisfying the inequalities O
=

J0

eetf(t)dt

and

F(x)

+oO Ill

Hence, (26) implies that the following A = det

= e5

determinant

+oO J z

.

(30)

vanishes:

eetfv(t)dt

k

emtf(t)dt

IIO
7

where v and lc are the row and column indices, respectively. If we successively subtract the first column from the zero-th, the second from the first, and so on, we reduce A to the form

where one puts +oo in place of m + 1 in the last column. We now introduce a different variable of integration in each column; we obtain:

1 xl) ...

1 Xl ...

*** ...

1 x, .

xo”

XT

‘*a

Xrnm

dxo..-dx,

=

$2. Hermite’s

.

fl

Method

87

(xj - xi) dxo . . . dx,

.

O
The integrand has constant sign in the region of integration; hence, A # 0. This contradiction proves the existence of f”(z) with the required properties. 5) ([Stieltjes 18901). Set f(x)

= xyx

- l)n+rl

. . . (x - my+r=

)

where the parameters ~1, . . . , r,,, will later be chosen equal to either 0 or 1. This polynomial clearly satisfies (20) and (22). From (21) and (24) we find that -(a,F(m)

+ 1.. + aoF(

= 2

ekak lk

emtf(t)dt

.

k=O

If we denote 1, 0, 1 can be rewritten ‘1’3k(x)

then the above equality a,F(m)

=

+ * *. + aoF

ifzsk; ifx>k, in the form

= -

J

Om evtf(t)G(t)dt

,

(31)

The function G(t) $0, and it is constant on where G(t) = CT=“=, e”ok6k(t). the intervals between integer points. We now choose the exponents ri , , . . , r, so that when G(t) and f(t) p ass through each of the integer points 1,. . . , m, they either both change sign or both keep the same sign. Then the integrand on the right in (31) has constant sign in the region of integration; this gives us (25). 6) ([Hilbert 18931). Set = xn(x - y+l

f(x)

. . . (x - my+1

.

Then f(x)

= d,xn

+ d,+lx

Since Jofbo e?‘dt F(0) Furthermore, F(k)

= ek

n+l

= r(r

+ - -a,

di E 25,

d,, = (-l)(“+l)m(m!)n+’

+ 1) = r! for r 1 0, it follows

= c d,r! f n!(-l)(“+i)“‘(m!)“+i T-272

.

that

(mod (n + l)!) .

for any integer k = 1,. . . , m we have (see (30))

i-m J

emtf(t)dt

k

=

J

o+m eetf(k

+ t)dt =

c r&z+1

c,.~!,

GEZ,

Chapter 2. Effective Constructions in Transcendental Number Theory

88

so that F(lc) G 0 (mod (n + l)!). Thus, a,F(m)

+. . . + aoF

E aoF

z ao,!(-l)(n+l)m(m!)n+’

(mod (n + l)!) .

There exist arbitrarily large n for which n + 1 is prime to aem!. For such n the right side of the last congruence is not divisible by (n + l)!, and so the left side is nonzero. This gives us the condition (25). Hurwitz proposed that n be chosen so that n + 1 = p, where p is a large prime number, p > ]as], p > m. This version of the proof that e is transcendental is probably the one that is best known. 2.3. The Lindemann and Lindemam-Weierstrass Theorems. As we mentioned before, in 1882 Lindemann solved an ancient problem by proving the impossibility of squaring the circle, i.e., by proving that x is transcendental. He actually obtained a much stronger result [Lindemann 18821. Theorem 2.2. If o is a nonzero algebraic number, then ea is transcendental. In particular, Q = 1.

Hermite’s Theorem 2.1 follows from Theorem 2.2 if we set

Corollary 2.2. All nonzero numbers that are logarithms of algebraic numbers are transcendental. In particular, x = i ln(-1) is transcendental. Using Euler’s formula eiz=~~~~+isinz, we easily conclude that if a E Cc, a # 0, then there is a transcendental number in each pair (a, sin a), (a, cosa), (a, tan a), (a, cot a), (a, arcsina), (a, arccos a), (a, arctan a), (a, arccot a). Although at first it might seem very strange, these results can be applied in theoretical mechanics. For example, in Lyapunov’s work ([1959], p. 100) he usesthe fact that arctan C is transcendental if C is a nonzero algebraic number. Lindemann stated without proof an even stronger result, and indicated that it could be proved using the same ideas. Theorem 2.3. numbers

If ~0, al,. . . , am

are distinct algebraic numbers, then the

eao,eal,...,eam are linearly independent over the field of algebraic numbers. In 1885 a proof of this theorem was published in [Weierstrass 18851. It is now customary to call this result the Lindemann-Weierstrass Theorem. Note that Theorem 2.2 is a corollary of Theorem 2.3, as we see by taking ra = 1, cro = 0 and al = a. It is not hard to show that Theorem 2.3 is equivalent to the following assertion:

52. Hermite’s

Method

89

Theorem 2.3’. If al,. . . , (Ye are algebraic numbers that are linearly independent over Q, then the numbers

eQ1,...,e

Q.

are algebraically independent over the field of algebraic numbers. This was the first theorem on algebraic independence of a set of numbers. The next such theorem did not appear until 1929 (see Chapter 5). We now give two other consequences of Theorem 2.3. Corollary 2.3. Let P(zo, . . . , z,) E A[zo, . . . , z,] be a nonzero polynomial with coeficients in the field of algebraic numbers that is not divisible by any non-constant polynomial in 20. If crl , . . . ,oS are algebraic numbers that are linearly independent over Q, and if p is a nonzero root of the equation P(z, eal’, . . . , ea.‘) = 0 , then /I is transcendental. Corollary 2.4. Let ~0, cl, ~2,. . . be a periodic sequenceof algebraic numbers. Then the value of the series f(z)

= gcn$ n=O

at any nonzero algebraic point z is transcendental. If the sequence {cn} is purely periodic of period m, then f(z) is a solution of the differential equation ycrn) - y = 0 with algebraic initial conditions. It is equal to a linear combination with algebraic coefficients of the functions ePb-Z7

p = exp($),

k=1,2

,...,

m.

The corollary in the purely periodic case now follows from Theorem 2.3. On the other hand, if the sequence {cn} becomesperiodic starting with Cd,then we can obviously add to f ( z ) a suitable polynomial of degree d - 1 with algebraic coefficients so as to obtain a function of the same type as in the corollary but with a purely periodic sequence of coefficients. This proves Corollary 2.4. In [1985] Ehrenpreis published another corollary of Theorem 2.3. Corollary 2.5. Let f(z) be a solution of an m-th order linear homogeneous differential equation with constant algebraic coeficients. If f(z) has m zeros that are algebraic numbers (counting multiplicity), then f(z) = 0. In the real case this is an immediate consequence of Rolle’s Theorem; but the situation is more complicated in the complex case. The proof of Theorem 2.3 given below essentially follows Lindemann and is based on a generalization of Hermite’s method. Instead of rational approximations to powers of e, Lindemann uses Hermite’s identity to construct approximations to algebraic numbers.

90

Chapter 2. Effective Constructions

2.4. Elimination of the Exponents. using Hermite’s argument in $2.1.

in Transcendental

Number Theory

The lemma that follows

can be proved

Lemma 2.1. Suppose that CZO,. . . , CE~, ao, . . . , a, are complex numbers, the function A(t) = aOeaot + . . . + a,eQmt satisfies form f(t)

the condition

A(1)

= 0. If the polynomials

- w) n+l . . . (t - &Jn+l,

= (t - ao)n(t

f(t)

and g(t)

have the

g(t) = $ C f(i)(t) . z>n

, (32)

where n is a natural

number,

where c is a positive not on n.

number that depends only on ~0,. . . , a,, ao, . . . , a,

Proof.

In analogy

with

and

then

and

(21), we have the relations

dO>eab J

oab emtf (t)dt,

k=O,...,m,

(34)

where the integration is taken over intervals with endpoints at 0 and ok. If we multiply (34) by ak for each k, add the resulting expressions, and use the condition A(1) = 0, we find that 2 k=O

akg(ak)

= 2

2

(akeab 1””

emtf (t)dt)

.

k=O

On the right we use the obvious bound msxltllr If (t)l 5 cy, where maxs
r =

In Hermite’s proof that e is transcendental, the left side of (33) turned out to be 2 1 in absolute value (by the choice of n), and this gave a contradiction for n sufficiently large. Below we shall show how to get an analogous situation in the case of the Lindemann-Weierstrass Theorem. Lemma 2.2. Let (~0,. . . , (r,,ae, . . . ,a, be algebraic numbers, and let A(t) $0 be the function in Lemma 2.1. If the Taylor series of A(t) near t = 0 has rational coeficients, then A(1) # 0. Proof. Let lK denote the finite extension of Q generated by ~0, . . . , Q,, ao,..., a, and all of their conjugates. Let v denote the degree of K, and let 01,9,... , gy denote all of the automorphisms of K. For every y E lK we let r(j) = uj (y) denote the image of 7 under the automorphism aj. If p(t) = -yo + y1t + -y2t2 + * * * is a power series with coefficients in K and ~7 is

52. Hermite’s an automorphism

of EC, then we let I

Method

91

denote the series with

coefficients

eYjYj)We make some remarks to simplify the argument. In the first place, without loss of generality we may assume that the coefficients aa, al, . . . , a, are algebraic integers. In the second place, we may suppose that the numbers oi are all distinct, since we could combine terms with the same exponential function without changing A(t). At least one of the coefficients aj is nonzero; we shall suppose that as # 0. Let d be a common denominator for the numbers oj, and set D = d”+‘. Then (32) implies that P+’ f(t) E ZZx[t] and P+ig(t) E Z&l. The degree of the polynomial g(t) is at most m(n + 1). Hence, I = D2(“+1)(u0g(a0)

+ aig(ai)

+ . . . + a,g(a,))

E 72~ .

(35)

For the polynomial ~ c f(i)@) h(t) = (n : l)! i>n+l we also have

J = D2(n+1)(aoh(ao)

+ alh(cq)

+ -. . + amh(cr,))

E 25~ .

We now choose a sufficiently large natural number n with the property that n + 1 is prime to d and to the norms of the nonzero algebraic integers d(as - oj), 1 5 j 5 m, and ac. We have

I = aoD2(n+1)I = a0 d2(“+l)

(

n!f

(n)(~g) + (n + 1) J

from which, by the choice of n, it follows We consider the functions

Aj(t)

= a$@t

+ (n + 1) J ,

(a0 - al) . -. (a0 - am))n+l

+ aIj)e+

that I # 0.

+ . . . + at.7&~:)t,

j=l,...,v.

It is easy to see that Aj(t) = cj(A(t)). But by the hypothesis of the lemma, the coefficients in the expansion of A(t) are rational numbers. This implies that all of the functions A,(t) have the same Taylor series at t = 0; hence,

A(t) = Al(t)

= . . . = A,(t) .

If we now suppose that A(1) = 0, we obtain Aj(1) By Lemma 2.1, this leads to the inequalities

Ia~‘gj(&))

+ alj)sj(a(j)) 1

= 0 for j = 1,2,. . . , V.

+ . . . + ag)gj(ag))I

where gj(t) = Uj(g(t)) an d cs is a positive n. According to (35), we now have

constant

n+l <- c2 72.I

’

that does not depend on

92

Chapter 2. Effective Constructions in Transcendental Number Theory

Since I is a nonzero algebraic integer, its norm N(I) nonzero. Using (36), we find that

belongs to Z and is

This is a contradiction for n sufficiently large, and Lemma 2.2 is proved. 2.5. End of the Proof of the Lindemaun-Weierstrass that the conditions in Theorem 2.3 are fulfilled, and agea + alea + . . . + u,ea”

Theorem. Suppose

= 0,

(37)

where ae,...,a, are algebraic numbers not all zero. This equality can be rewritten in the form A(1) = 0, where A(t) is the function in Lemma 2.1. By assumption, the numbers oj, 0 5 j 5 m, are all distinct, so that the function A(t) is not identically zero. In what follows we use the notation introduced in the proof of Lemma 2.2. Consider the function ”

B(t) = n

Aj(t)

= boePot+ bleolt + . . . + bMeflMt ,

j=l

which is not identically zero. All of the bj and bj lie in lK. The Taylor coefficients of B(t) at t = 0 also lie in K, and if we set B(t) = CrZoynyntn, let g be any automorphism of K and let r be the permutation of the indices {1,2,. . . > V} for which (T(T(= (T,(~J, then we have ”

co

”

C c(~n>t” =n A,(j)(t) =n Aj(t>=B(t). n=O j=l

j=l

Thus, all of the conjugates of y,, are the same, and so the coefficients ^ln are rational numbers, i.e., B(t) satisfies the conditions of Lemma 2.2. Since A(1) = 0, we have B(1) = 0, contradicting Lemma 2.2. This completes the proof of the Lindemann-Weierstrass Theorem. 2.6. Generalization of Hermite’s Identity. The identity (18) may be viewed as a consequence of the fact the eZ is a solution of the differential equation y’ - y = 0. In [1883], H urwitz proved an analogous identity for any function Y(x) that satisfies a differential equation of the form axy ” = by’ + y,

a,bcC.

(38)

He showed that for any polynomials g(x) and h(x) one can find polynomials gi(x) and hi(x) such that

53. Functional Approximations

g(z)Y(z)+ h(W(4 = g (!ww From this we obtain

J

(g(z)Y(z)

Hurwitz

which

the following +

used this identity

satisfies

Theorem

f’(c)/f(C)

+ h(W(z)) .

analogue of Hermite’s

h(+-+))d~

=

91 bP-(z)

to prove the following

the differential

93

equation

+

identity: hl

theorem

bJ)Y’(x)

*

for the function

(38).

2.4. Let d E N, IK = Q(a), does not lie in K.

C E RI, < # 0. Then the number

Hurwitz’s argument can easily be generalized to functions that satisfy differential equations of the form

2 Pk(2)y(k) = 0,

PO,. . . , Pm E ZK[Xl *

k=O

This was done by E. Ratner and by Hurwitz himself. As a result, in [1888] Hurwitz proved Theorem

2.5. Let P(z), Q(x) E Z,[z],

n = deg Q > deg P,

(39) Then for any nonzero C E R at least one of the numbers f’“‘KMC),

s=l,...,n-1,

does not lie in UC Much stronger results have been proved using a method that was proposed by Siegel in 1929. We shall describe these results in Chapter 5.

$3. Functional

Approximations

3.1. Hermite’s Functional Approximation for ex. Hermite’s construction of simultaneous rational approximations to powers of e is based on a certain analogy between numerical and functional simultaneous approximations. The numbers e, e2,. . . , em may be regarded as the values at z = 1 of the functions eE,e2Z, . . . , em’; and the first thing that Hermite does is to construct “good” approximations for these functions using rational functions

94

Chapter 2. Effective Constructions

in Transcendental

Number Theory

in z all having

the same denominator. In other words, he finds polynomials of bounded degree such that as many initial terms as possible vanish in the Taylor series at z = 0 for the function Qs(z)e”* - Qk(z), k = l,... ,m. Suppose that we consider a function f(z) to be “small” when the first nonzero term in its Taylor series at z = 0 has a large power of z. If we want to find polynomials &i(z) with given restrictions on their degrees so that the functions Qo(z)ekZ - Qk(z), k = 1,. . . ,m, are all as “small” as possible, one possible approach is to solve a system of linear equations in the unknown coefficients of the desired polynomials. This system is obtained by setting the initial coefficients of the Taylor series equal to zero. Hermite, however, found a more explicit way to construct the polynomials. be distinct complex numbers, let no, nl , . . . , n, be nonLetai,crz,...,a, negative integers, and set

&o(z),Q~(zL~~~, Qm(z)

f(x) Setting

= Zno(Z - al)nl

M = no + . . . + n,,

. . . (z - ct,p

.

we have the identity ab

Qo(z)eabz- Q&Z)

= z

M+le(2bJ

I

where deg &j(z) 5 M - nj. mite identity that was used Weierstrass theorem. In 1893, Hermite studied (see [Hermite 19171, Vol. 4, tion of approximations that complex plane that contains

k=l,...,m,

emztf(t)dt, 0

(40)

If we substitute z = 1 in (40), we obtain the Herto prove transcendence of e and the Lindemannanother identity involving exponential functions p. 357-377), which in some sense gives a construcis dual to his earlier one. Let C be a circle in the ~1,. . . , om E UZ. Then . . . (C

-

(y,p+1

(41) = 2

Pk(z)eabz

,

k=l

where Pk(z) is a polynomial of degree nk in z, 1 <_ k 5 m. If we compute the residue of the integrand at infinity, we easily see that ord,,cR(z) = -1 + Cr=“=,(nk: + 1). Here and for the rest of this section ord,=,f(z) denotes the multiplicity of zero of f(z) at the point z = cr. Hermite actually found a similar identity as early as 1873. In the case - 0 he published the following representation for the remainder in an rFpr;ximation (see [Hermite 19171, Vol. 3, p. 146-149): 1

R(z)

=zN

1

J J ff.

0

where N = -1+

cp=“=,(nk

0

+ l),

WZl,..

* 7 &n-l)'

$3. Functional ,X,-l)

V(Xl,... Ubl,

*

. . ,X,-l)

Approximations

95

=(a1 - cQ)Xl *. . xm-1 + . . * . . . + (wn-2 - %n-&,-25??&-1+%%-1Gn-1

,

=(l - xi)nz . . . (1 - Z,-i)n,. nl+...+n,-~+m-2 . xnlxnl+nz+l~~ 1

2

'%n-1

If the cri are algebraic numbers, then R(1) = 2

Pk(I).+

kc1

is a linear form in the numbers eak with algebraic coefficients. The form is rather small as n or m increases in absolute value. Such a sequence of linear forms can be used to prove arithmetic results about the values of the exponential function. 3.2. Continued Fraction for the Gauss Hypergeometric F’unction and Pad6 Approximations. The continued fraction expansion of a number is closely connected with rational approximations of that number (see 81.4 of Chapter 1). But unfortunately, the continued fractions are known for very few real numbers. They are not even known for basic constants such as 7r; and computer calculations have not revealed any regularity in the continued fraction expansion of 7r. On the other hand, since the time of Gauss, functional continued fractions have been found for many classical functions. These continued fractions are irregular, i.e., the numerators may be different from 1. They can be used to construct rational approximations to the values of the functions, and hence to prove estimates for the irrationality measure of these values. In particular, in this way one can obtain results on approximating r. Standard facts from the general theory of continued fractions can be found, for example, in [Perron 1954, 19571. Let a,b,c E Cc, c # 0, -1, -2,. . .. The Gauss hypergeometric function is defined by the power series

(a)v@L ” , F(z) = F(a, b,c; z) = cO” -z “=O (CW where (o)s = 1 and (Q)~ = cr(o + 1). .. (a + v - 1). If a or b is in the set {O,-l,-2,. . .}, then F( z ) is a polynomial; otherwise, it has radius of convergence 1. By making special choices of the parameters a, b, c, one can obtain many classical functions, for example, ln(1 + z) = zF(1, 1,2; -z), Arcsinz = zF(1/2,1/2,3/2;

Arctanz z2),

= zF(1/2,1,3/2;

(1 - z>r = F(-r,

Gauss found the following continued fraction expansion:

-z2) , 1,l; z) .

Chapter 2. Effective Constructions in Transcendental Number Theory

96

F(a, b, c; z) F(a, b + 1, c + 1; z)

=1+

al4 + -a24,I

- ,I

a34

+J1+)1+“‘,

a44

(43)

where the complex numbers a, are given by the formulas (a + k)(c - b + k) a2k+1 = - (c + 2k)(c + 2k + 1) ’

(b + k)(c - a + k) a2k = - (c + 2k - l)(c + 2k) ’

This continued fraction converges to the function F(a, b, c; z)/F(a, b + 1, c + 1; z) in the region 0 < arg(z - 1) < 27r (see 521 of [Perron 19571). The numerator An(z) and the denominator B,(z) of the convergents in (43) are polynomials with complex coefficients, and

degh

I

[f 1,

degA,

5

[ 1 n+l 2

.

We have the identity B,(z)F(a,

b, c; z) - A,(z)F(a,

b + 1, c + 1; z) = (-l)nai

. . . a,+izn+‘fn+2(z)

,

where fik(Z) = F(a+k,

b+k, c+2k; z),

f2k+l(z)

= F(a+k,

b+k+l,

c+2k+l;

z) .

It was Riemann (see [1876], p. 400) who first proved the existence of such a relation, based on considerations from the analytic theory of differential equations. By expressing the right side as an integral and computing its asymptotic behavior as n + +oo, he studied the question of when the convergents approach the corresponding hypergeometric functions in the limit. The meaning of the above relations is that the numerators and denominators of the convergents in (43) give good approximations to the ratio F(a, b, c; ~)/&‘(a, b + 1, c + 1; z) in the functional sense. We have corresponding results for the function F(a, 1, c; z). That is, (43) implies the following continued fraction expansion:

F(a,l,c;z)=~+ 11 ---Gzl Il + 44 Il + @I Il + 44 , 1 +..* 7 where af is obtained from ai by substituting b = 0 and replacing c by c - 1 in (43). In this case we can give explicit expressions for the denominators of the convergents: B;,(z) B,*,+,(z)

= F(-n, = F(-n,

1 - a - n, 2 - c - 2n; z) ,

-a - n, 1 - c - 2n; z) .

These formulas hold for n 2 0. Actually, similar formulas for the numerators and denominators of the convergents are known in the general case of Gauss continued fractions (see [Heine 18571and [Heine 18781, p. 281), but they are more complicated.

$3. Functional Approximations

97

In [1907] Pade (who was a student of Hermite) found explicit expressions for rational approximations to F(a, 1, c; z) with various restrictions on the degrees of the numerators and denominators of the convergents (see [Pade 19071,p. 389). Jacobi found the following useful representations for the remainder and the denominator of the convergents to the function F(a, 1, c; l/z) in a neighborhood of infinity (see [Jacobi 18911, p. 196). Let Rea > 0, Re(c - a) > 0; let Q,(x) = F(-n, c + n - 1, a; Z) &a(1

-5)l+a-c (u),

=

d .

72

z (

(xa+n--l(l

-

x)c-a+n-l)

)

be the Jacobi polynomial, deg Qn(z) = n; and let the degree n - 1 polynomial P,(z) be defined as

s 1

ta-~(l

P,(x)

=

_

t)c--a--l

Qnct; I

,&&I

dt

.

0

We then have the following Jacobi

identity: 1 p+n-l(l-t)c-a+n--l

xPn(x)

- Qn(x)F(u,

1, c; l/z)

= $-

1 n 0

(t

-

++1

dt

*

We note that the Taylor series for the right side in a neighborhood of infinity begins with the term yx-“. In addition, B;,(x)

= (n +’ - l)nx”Q n!

n

(l/x)

.

In recent years this identity has often been applied when proving bounds on the irrationality measure of various numbers. Using the continued fraction expansion (43)) one can obtain continued fractions for the so-called conflzlent functions (see 524 of [Perron 19571). For example,

$‘(c;z) ~(c+I;z)=l+]c+l

z/c I

ZI

+1c+2

ZI

+1c+3

ZI

+1c+4

+“”

where

In these casesone can also write out explicit formulas for the remainders and for the numerators and denominators of the convergents (see [Rossum 19551). The above functional approximations have been used in many places (starting in 1908 by Thue; see $3.2 of Chapter 1) to obtain results on rational approximations of the values of the hypergeometric function and its specializations (see §5), and also to investigate certain Diophantine equations. It is

98

Chapter

2. Effective

Constructions

in Transcendental

Number

Theory

notable that in almost all cases the proof of the required functional relation is immediate. Hermite-Pad4 Functional Approximations. Suppose that m > 2, , fm(z) are functions that are analytic at z = 0, and ni, . . . , n, are h(z),... nonnegative integers. As we saw in our earlier examples, one sometimes needs to find effective solutions to the following two problems: 3.3. The

1)

find a set of polynomials Pi (z), . . . ,P,(z) tions degPi(z)

2)

5 ni - 1,

E C[z] that satisfy the condi-

ord,=o (gE(z)fi(z))

> -I+gni;

find a set of polynomials Al (z), . . . , Am(z) E C[z] that satisfy the conditions deg Ai

5 rs - ni,

or&=o(-h(z)f~(z)

- Aj(z)fi(z))

2 CT+ 1

for 1 5 i < j 5 m, where 0 = Cc”=, ni. By comparing the number of coefficients of the polynomials with the number of conditions imposed on them, it is easy to check that the desired polynomials exist for any choice of ni , . . . , n,. The m-tuple of polynomials Pi(z) is called a Hermite-Pade’ approximation of the first kind for the functions and the polynomials Ai are called a Hemite-Pad& apfl(z),~~~>fm(z); proximation of the second kind. In §§3.1-3.2 we saw examples of such approximations of both the first and second kinds. The sections that follow will describe other examples and their application to proving number-theoretic results. In the mid-1930’s Mahler studied the general properties of these approximations and, in particular, the relationship between approximations of the first and second kinds. However, his manuscript on the subject was not published until 1968 (see [Mahler 196Sa]). Mahler’s theory was further developed in [Jager 19641and [Coates 19661, whose authors had read Mahler’s work before its publication. Here we cannot dwell in more detail on the results of this theory, so the interested reader will have to consult the papers cited above. In proofs of number-theoretic results - estimates of the irrationality measure of numbers and of linear forms in certain numbers - one often finds that what appear to be completely different functional constructions actually give the same system of functional approximations, up to a constant factor.

$4. Applications

of Hermite’s

Simultaneous Functional Approximations

99

94. Applications of Hermite’s Simultaneous Functional Approximations We shall not dwell on the many variants of the proof of transcendence of e and K, or of the Lindemann-Weierstrass theorem. We gave some idea of this in 31. For references to the extensive literature on the subject, see the survey [Fel’dman, Shidlovskii 19671 or Mahler’s book [1976]. It should be noted that new variants on the classical proofs have continued to appear in recent years (see, for example, [Nesterenko 19871, [Beukers, Bezivin 19901, [Bezivin, Robba 19891). 4.1. Estimates of the Transcendence Measure of e. The transcendence measure (see the Introduction) of e can be estimated using a refinement of Hermite’s method. We show how this is done. Let

P(z) = a,xm

+ -. * + alx + a0 E if@],

am # 0.

In $2 we used Hermite’s identity to prove that there are infinitely many (m+l)tuples of integers (bs, . . . , bm) for which a) b)

A=a&+--+ am& is a nonzero integer; the numbers &k = beck - bk, Ic = 1,. . . , m, are so small that

From the equality

b@(e) = a&o + . . . + am&

+ al&l + - - ++ amEm

we find that

lb&e)1

2 Ia&0 +. . . + am&l

- /alEI + * *. + amhI

1

> - ; 2

in other words, IP(

2

@bo)-l

.

(44

Of course, among all (m + 1)-tuples (be, . . . , bm) satisfying a) and b) we have to choose one for which ]be] is as small as possible, so that the above bound on [P(e)] is as good as possible. This value of Ibe] increases along with H = H(P) = maxe
> H-C’l”‘nH

,

where c is a positive constant that depends only on m. This result was actually the first bound for the transcendence measure of any number. In [1927]

100

Chapter 2. Effective Constructions

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Number Theory

Mordukhai-Boltovskoi obtained bounds for the transcendence measures and Inca for (Y E A. He also found a lower bound for the linear form me 01 + . . .+ateBt,

al ,...,

at,pl,...,

of 7r

/3t EA.

These early quantitative results were still far from best possible. In [Popken 19291 Borel’s result was strengthened as follows. Theorem 2.6. For any P(Z) E Z[Z], deg P = d, P $0, greater than a certain bound depending on d, then [P(e)1 where

q,(d)

is a positive

constant

> Hmd-* depending

H(P)

= H,

if H is

, only on d.

If we compare the inequality in this theorem with the upper bound in (6) of the Introduction, we see that the factor Hmd on the right is correct, i.e., the -d cannot be replaced by a larger exponent that depends only on d. Popken’s proof was a quantitative version of Hermite’s method. Like Borel, he used a variant of the Hilbert-Hurwitz argument, in which the prime p = n + 1 is chosen prime to asm! (see 51.4). Popken found a way to remove the condition p > ]ae], which had forced Bore1 to choose p sufficiently large. Popken assumed that the coefficients as, al,. . . , a, of the polynomial are relatively prime integers, and chosethe polynomial f(z) in Hilbert’s argument in the form f(z) = (LC- r)-l(z(2 - 1) a.. (z - m))n+l , where r is the index of a coefficient a, that is not divisible by p. In this way he could ensure that condition (8) in $1 holds for p > m. Popken’s result was refined in [1932a] by Mahler, who also proved Theorem 2.6 and showed that one can take Q(d) = cd2 In d, where c is an absolute constant. Mahler used a quantitative version of Hermite’s argument (see 52.2 of Chapter l), and to get condition (8) he provided a proof that the set of simultaneous approximations is linearly independent, i.e., the equality (26) is impossible. 4.2. Transcendence of e”. It seemsthat the first application of the identity (41) in transcendental number theory was connected with the proof that e” is transcendental. This fact, which is a special case of Hilbert’s seventh problem (see Chapter 3), was proved in 1929 by Gel’fond. In (41) let us set 121 = ... = 12, = 0. We enumerate the elements of Z[i] in the order of increasing absolute value (and increasing argument when the absolute value is the same), obtaining the sequence (~1,(~2,. . .. Let Rm(x) denote the left side of (41) with .z = X. Prom the expression on the right we find that Rm(r) E Q[eA, emK, i] for any m E N. Let r = lo,] = O(m’i2). Gel’fond showed that for any m E N there exists R, E N satisfying the conditions

54. Applications

of Hermite’s

Simultaneous Functional Approximations

101

1) 0, .R,(r) = Pm(e”,e-“,i),P,(TY,z) E W,Y,~, 2) 3) 4)

lnL(P,) = O(r2), degP, < r, IPm(en, emK, i)l 5 exp(-rrr2

lnr + O(r2)).

The rest of the proof that en is transcendental is based on the argument that we described in $1.5. If we suppose that ek is algebraic, we see that for R,(r) # 0 and for m sufficiently large, the inequality in 4) is incompatible with the lower bound for IPm(err, e-=, i)] that one obtains from l), 2), and 3) using Theorem 1.5. Hence, &(?r) = 0 for m 2 mo. If we now show that some of the R,(n) with arbitrarily large m are nonzero, then we will have a contradiction, and we will have proved that e” is transcendental. Here Gel’fond used the fact that the numbers Rm(r) are the copoints efficients of the expansion of eKZ in a Newton series with interpolation al,cQ,. . ., In other words, for any complex z we have the identity

infinitely many of the coefficients must be Since elrz is not a polynomial, nonzero. In his paper Gel’fond noted that one can prove transcendence of oiO in exactly the same way, where cx is an algebraic number not equal to 0 or 1 and D is a positive integer that is not a perfect square. (Note that en = (-1)-i.) In [1930] Kuz’min showed that the same method can also be used to prove transcendence of crJzT, where D is a positive integer that is not a perfect square. In a similar way Boehle [1933] proved that if cr is an algebraic number not equal to 0 or 1 and /? is an algebraic number of degree d 2 2, then at least one of the numbers 2, CP,..., a@-’ (45) is transcendental. Using the same method, he also proved a theorem on simultaneous approximation of the numbers (45) by rational numbers. The same approach enabled Siegel [1932] to prove that at least one of the periods of a Weierstrass elliptic function with algebraic invariants must be transcendental. Schneider later proved that, in fact, both periods are transcendental; but he used a different method. We shall discuss this further in Chapter 3. Using the same technique, one can prove results about transcendence of eQ for nonzero algebraic (Y (see [Gel’fond 19731). In this case one must take ak = k - 1, k = 1,. . . , m, where m = 1 + dega, and take the ni so that ni 5 7x2 5 ... < n, 5 ni + 1. Then the integrals RN(Q), N = ni +. . . + n,, again turn out to be the Newton series coefficients for ear, but with interpolation points O,l,..., m - 1.

102

Chapter 2. Effective Constructions

in Transcendental

In [1932] Koksma and Popken proved the following of Gel’fond’s theorem.

Number Theory

quantitative

refinement

Theorem 2.7. Let E be an arbitrary positive number. Every polynomial P E Z[z], P(z) f 0, satisfies the inequality

where H = max(H(P),3) deg P and E.

and C is a positive constant that depends only on

Koksma and Popken used Gel’fond’s analytic construction, and proved that there exists a rather dense sequenceof coefficients Rm(r) in the above Newton series for err= that not only are nonzero, but also have a good lower bound depending on the index m. As a consequence they found that for any X in the interval 1 < X < 2 and for all h E N sufficiently large, there exist n with h 5 n 5 Xh and a polynomial A,(z) E Z[i][z] such that deg& 2x - 1 --nlnn 2x

L W%

H(A,)

5 ecln ,

(46)

5 exp . - csn 5 IAn( (47) > > The proof of Theorem 2.7 uses this sequence of polynomials and proceeds in two stages. In the first stage one supposes that the polynomial P(x) is irreducible. The parameter n is chosen in a special way as a function of H. If the polynomial A,(z) is divisible by P(s), then the lower bound in (47) gives’ a lower bound for IP( in terms of n. Since we know how n depends on H, this can be converted to a lower bound in terms of H. If A,(z) is not divisible by P(z), in which case the two polynomials are relatively prime, one can use their resultant to obtain a lower bound for max(]Ai,(e*)], IP(e” and then for IP( as well. Later (see Lemma 6.1) we shall discuss this fact from elimination theory in more detail in connection with algebraic independence results. In the second stage of the proof P(x) is an arbitrary polynomial with integer coefficients. If we write it as a product of irreducible polynomials and apply the result proved in the first stage to each factor, we obtain a lower bound for IP( in terms of the heights of the irreducible factors. In order to express this bound in terms of the height of P(z), Koksma and Popken used a relation between the heights of a set of polynomials and the height of their product. Similar considerations (expressed somewhat differently) were used by Mahler in 1932 to prove bounds for the transcendence measures of x and the logarithms of rational numbers (see s4.4). In 1948, using a similar argument and taking into account the variation in the degreesof the polynomials, Gel’fond proved a criterion for numbers to be algebraically independent (see Lemma 6.4) and obtained some important results on the transcendence degree of fields generated by values of the exponential function. ew

$4. Applications

of Hermite’s

Simultaneous

Functional

Approximations

103

4.3. Quantitative Refinement of the Lindemannweierstrass Theorem. The following result of Mahler [1932a] significantly strengthens the MordukhaiBoltovskoi theorem on linear forms and polynomials in the values of the exponential function at algebraic points. Theorem 2.8. Let 01, . . . ,8, be algebraic numbers that are linearly independent over Q. There exists a positive constant c depending only on the Bi such that for any polynomial P E Z[xl, . . . , x,.], di = deg,, P, one has (P(e*‘, . . . , e&)l 2 H--c&-& for any H > H(P) and also ondl,...,d,.

that exceeds a certain bound that depends on el, . . . , &

We note that when all & E Iw the constant c cannot be less than 1; otherwise, it cannot be less than l/2 (see (7) in the Introduction). The algebraic part of the proof of this theorem makes use of a quantitative version of Siegel’s argument in 51.4 (see also Chapter 5). The required linear forms that take small values at 3 = (emi,. . . ,ecrm), CQE A, are constructed using the identity (41), which gives simultaneous functional approximations for ealr,...,eamz. In order to get a full system of such forms, Mahler used a technique that goes back to Hermite (see [Hermite 19171,Vol. 4, p. 368). For large n E N and for each i = 1, . . . , m we choose nk in (41) as follows: nk=n-1

if k#i;

ni=n.

Let Ri(z) and Pi,k (z) denote the function R(z) and the polynomial Pk(z) corresponding to this choice. Then Ri(z) = $J Pi,k(Z)eaLZ,

i = l,...,m.

k=l

For any i we have N = mn (see (42)). It is easy to compute the determinant A(Z)

=

det

IIPi,k(z)lll
.

Namely, from the construction of Pi,k(z) it follows that A(Z) = Azmn + Bzmn--l + . . . , where A = (n!)-”

m n k=l

n l
(ak - Cyi)-n # 0 . i#k

On the other hand, since ord,,oRi(z) 1 N = mn, it easily follows that ord,,,-,A(z) > mn. Thus, A(Z) = Azmn, and A(1) = A # 0. We now take the crj to be algebraic numbers, and set

104

Chapter 2. Effective Constructions

in Transcendental

L,(Z) = n!qN2 Pi,k(l)Xk,

Number Theory

i=l,...,m,

k=l

where q E Z, q # 0, is such that q(cxl - ctj)-’ E ZA, 1 5 i < j 5 m. It follows from the above that these linear forms Lo are linearly independent. It can be shown that the Li(Z) have coefficients that are integers of the field !K = Q(a1,..., a,), and that these linear forms take rather small values at the point < = (eal , . . . , earn ). It is to these linear forms that one applies the argument in 51.4. In particular, taking K = Q, one can prove Theorem 2.6 in this way. But in the case of an arbitrary algebraic number field K these considerations are not sufficient to obtain a lower bound for the absolute value of a single linear form at the point T, since our linear forms do not have coefficients in Z (see $1.4). Nevertheless, the argument in 51.4 can be modified (see Chapter 5) so that in this case as well, one can obtain the bound in Theorem 2.8 for the polynomials by regarding polynomials in eel,. . . , es7 as linear forms in the products of powers of these numbers and applying the above constructions to ok = s,e, + ‘*’ + s,.B,., si E No. This idea of Siegel will be described in more detail in $5.2 of Chapter 5 in connection with bounds for the algebraic independence measure of the values of E-functions. 4.4. Bounds braic Number.

for the Transcendence

Measure

of the Logarithm

of an Alge-

In the same paper [1932a], Mahler used the functional identity (41) to prove a bound for the transcendence measure of x and the transcendence measure of the logarithm of a rational number.

Theorem 2.9. Let w = 7r or else w = ln(a/b) E R, where a,b E N. Then there exist constants c = ~(a, b) > 0 and y = y(a,b,d) > 0 such that the following inequality holds for any P(x) E Z[x], P $0, deg P 5 d, H(P) 5 H:

IP(

2 ye H-Cd .

(48)

The algebraic cornerstone of the proof is a general fact about lower bounds for polynomials P(z) E Z[x] at a point w E Cc.In order to apply this fact, one has to show that there exists a sequence of polynomials P,(x) E Z[z] of fixed degree such that 0< H(P,)


C-Xlm 5 IPm(w)I 5 C-Xzm )

(4%

where c, Xi, and As are constants, c > 1, Xi > X2 > 0. The method of proof in essenceresembles the above argument of Koksma and Popken that they used to derive a bound for the transcendence measure of en. Mahler constructs the required sequence of polynomials using the functional identity (41) with ni = ... = nm = n and ok = /c - 1. If 8 # 1 is a positive rational number, then for fixed n and increasing m the sequence of expressions in (41) with z = In6 will be a sequence of polynomials of degree

$4. Applications

of Hermite’s

Simultaneous Functional Approximations

105

n in z = lni3 with rational coefficients. The bounds (49) can be proved for these polynomials. Here we emphasize that, unlike before, one works with increasing m and fixed n. To prove the two-sided estimate for this sequence of polynomials, for positive real z Mahler represents the integral 1 R(z, m, n) = : 27r2 c C”+l(C

J

e”CdC - l)“+l . - - (5 - m)n+l

(the left side of (41)) in the form of a multiple real integral, and then applies the mean value theorem to it. As a result he obtains the bounds ZN < R(z,m,n) XT-

ZN

5 emXF

,

(51)

where N = (m + l)(n + 1) - 1. To estimate the transcendence measure of r one has to consider the sequence R(27ri, m, n) of polynomials in n. In this case the mean value theorem does not apply. Mahler finds the asymptotic behavior of R(27ri, m, n) as m + 00, and again obtains the two-sided estimate (49). Mahler later extended the result (48) to the logarithms of arbitrary algebraic numbers Q # 0 (see [Mahler 1953b]). H e a 1so obtained explicit formulas for the constants c and 7. Theorem 2.10. Suppose that (Y E A, Q In CY# 0, In (Y denotes the principal value of the logarithm, v = dega, h = H(a), P(x) E Z[z], d = deg P, and H = H(P) 2 1. If T = max( [e4dv+1ln(v + l)], 50~ - l), and N = max(30 ln(T + l), [(ln H)/n] + l), then

]P(lncr)] 2 i(e2dNH)-(T+1)Y

.

(52)

Note that for fixed CYand for H 2 exp(c,#) and GJ= cc(o) the inequality (52) turns into (48). We give a few more inequalities of Mahler [1967]. All of them can be deduced by similar techniques. Let a, b, p, q, qk, ak E N. Then 12a- ebl > 23 - e2 = 0.6109.. + , lea - bl 2 u-‘~~,

IP(

> e&,

P E Z[z],

P $0,

a L a0 ,

deg P = d > do(H(P)),

(53)

c > 1 . (54)

In [1964a] Baker, while still using Mahler’s construction, was able to refine Theorem 2.9. Under some additional conditions on the positive integers a and b in w = ln(a/b), he proved that the right side of (48) can be replaced by where E is an arbitrary positive number and y depends on a, b, d, 7H -‘-‘, and E. Baker was able to strengthen Mahler’s result by regarding P(w) not

106

Chapter 2. Effective Constructions

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Number Theory

as a polynomial in w, but rather as a linear form in 1, w, . . . , wd with integer coefficients. The paper [Baker 1964a] contains several results on rational approximation of ln( 1 + i) for integers a > 0 that are obtained using the same functional construction. For example, it is proved that for any rational number p/q one has

I I ln2

-

f

2

In [Reyssat 19831 the same method tions of logarithms of rational numbers on the degree and height. Among the has: p(ln 2) 5 4.63, p(ln 3) 5 14.7 (for nent p(w) see the beginning of §4.5),

10-106~-12.5

.

(55)

is used to find a bound for approximaby algebraic numbers with restrictions concrete consequences of this result one the definition of the irrationality expoand also the inequality

) In 2 - H-lo5

,

which holds for all quadratic irrationalities t of height at most H, where H 2 Ho. We note that Gel’fond’s method has been used to obtain stronger inequalities than (48), (52), or (54) in the case of increasing degree of the approximating algebraic number or increasing degree of the polynomial being bounded. We shall discuss this in Chapter 3. 4.5. Bounds for the formulating theorems nient to use the notion This is defined as the the inequality

Irrationality Measure of m and Other Numbers. When about the irrationality measure of numbers, it is conveof the irrationality exponent p(w) of a real number w. greatest lower bound of the set of numbers p for which

has only finitely many rational solutions p/q. We have p(w) number w (see $1.4 of Chapter 1). In [1953a] Mahler proved the inequality

I I .I,-?

4

24-42,

> 2 for any real

q>2.

(56)

For q sufficiently large, the exponent 42 can be replaced by 30, i.e., one has b(r) 5 30. To prove this, Mahler uses the identity

F(x)

=

&

J c

p+l(c‘

= eAj(x)(lnx)j j=o

_

q.,+?

. . . (C -

m

+

lp‘+l

=

(57) ,

$4. Applications

of Hermite’s

Simultaneous

Functional

Approximations

107

n = max(nj), Aj(x) E (f&c], degAj(z) 2 m - 1. This relation is obtained from (41) with ok = k - 1 and z = lnx for any z # 0. It is used to construct a complete system of linearly independent functional linear forms for the set of functions (In z)j, j = 0,. . . , m; this is done along the lines of the proof of Theorem 2.8. Mahler chooses m = 10, x = i, lnx = 7r/2. Then for any rational number p/q one has a sequence of polynomials P,(y) E Z[i][y] such that where

deg Pn(y) 5 m, H(Pn) 5 Get;,

Pn(Pld

# 07

IPn(~)l I YOnY1^(zn 7

where the cj and ^/j are certain explicit constants greater than 1. A lower bound for 7~- E for large q (namely, for q 2 2.14 x 1014) is I I obtained from the inequality Q--m I IPnb/q)J

5 IPn(~)l+

IPn(r>

- Pn(Pldl

7

if we take n to be the smallest integer for which IPn(~T)I I2-‘qand take into account that

IPn(~>-

P7z(plq)l =

IA - bldl . Icx~)I

for some 8 between z and p/q. Any solution to the inequality

is a convergent of the continued fraction for X. But starting with the 13-th convergent, the denominators are greater than 2.14 x 1014. After computing the first 13 convergents of the continued fraction expansion of r, we seethat (56) also holds for all fractions with small denominators. In [1953b] Mahler proved the following theorem by the same method, except that he incorporated Liouville’s theorem to find a lower bound for the polynomials at an algebraic point. Theorem 2.11. Let Q E A, and let K = Q if CYE JRand IK = Q(i) if cr @IR. Suppose that P(z) E Z p<[z ] 2sirreducible over IK, P(Q) = 0, d = deg P. If m = 120. 25(d-‘)/2

II

H = max(H(P),

(m + l)(“+‘)ld)

,

then (58) In particular,

(58) implies that for nonzero (Y E A the series

108

Chapter 2. Effective Constructions

in Transcendental

03Zk ck=l sin&4

has radius of convergence

1, and the Dirichlet

z

Number Theory

series

ks sii(lco)

converges for Re s > d(m + 1) ln(m + l), where m and d are the integers in Theorem 2.11. In [1974] Mignotte, by choosing m = 5, was able to refine Mahler’s argument and prove that CL(~) < 20. With m = 10 and x = -1 he constructed a sequence of polynomials of degree 5 5 that have the required properties at the point 7r2, and used this construction to prove that p(.rr2) < 17.8. Another result in the same paper was the inequality ]]en]] > n-18n, which is stronger than (53) and holds for all n sufficiently large. Here ]I . I] denotes the distance to the nearest integer. In [1978] Danilov used the identity (41) to estimate the irrationality exponent of certain numbers. He proved that if a,b E z,

b>a>O,

(vwzi)2<~,

then p(ln(l

- a/b)) I

2 ln(&

- 6)

- ln(4b - 2a)

1+2ln(&-6)

’

A similar result was proved for the number dwarcsin(a/b). paper also contained the first estimate for the irrationality namely, &r/A) 5 9.35 .

exponent

(59) The same of ~/a, (60)

The proof of (59) was essentially based on the identity (41) with ni = . . . = n m= 1 and ok = k + 1. The following estimate for the corresponding &(z), z > 0, was obtained using the Laplace transform: 22m+l

(2m + l)!

e-o.5z(1

- e-“.5z)2m+1

2 R,(z)

5 c2~m~~l,

(1 -

e--0.5y+l

.

For z = - ln(1 - a/b) these bounds lead to a proof of (59). In [1982a] Chudnovsky used the inverse Laplace transform to obtain asymptotically precise upper and lower bounds for the integral R(z, m, n) (see (50)), for the coefficients Pj(e”, m, n) in the representation R(z,m,n)

= ~Pj(e*,m,n)Zj j=O

$4. Applications

of Hermite’s

Simultaneous

Functional

Approximations

109

for fixed z and increasing m and n, and for the common denominators (as a function of m and n) of the coefficients of the polynomials Pj (x, m, n). These

estimates were proved for positive real z and for complex z = iB, 0 < 0 < r. Chudnovsky used these results to prove bounds on the irrationality exponent of the natural logarithms of rational numbers. Taking n = 1 (Danilov’s case) and z = i7r/3, Chudnovsky proved that

I l- Id2 + a1 + ’ = 8 30998634

p(4)

ln(2 - &)

- 1

’

“.

’

(61)

and taking n = 5 and z = i7r/2, he proved that

The bounds in [Chudnovsky 1982a] for the irrationality on the following useful fact.

measures are based

Proposition 2.1. Let k be a natural number, and let w be a real number. Suppose that there exists a sequence of polynomials PN(x) E Z[x], N > No, such that deg PN < k and

Then

the irrationality

exponent

of w is no greater

than k(1 + C/T).

In [1993a] Hata significantly strengthened (62): he proved that p(r) 5 13.398 .

(63)

To estimate the irrationality exponent of K, it turned out that it is better to choose different powers of (C - k) in the integrand in (50), according to a certain principle. Namely, in [Hata 1993a] the denominator in (50) is replaced by QN+&)QN+~(z - r)QN--T(Z - 2r)Qiv--3r(Z - 3r) , where QN(z) = z(z - 1) ... (z - N) and r = [3N/31]. With this choice of integrand, in Proposition 2.1 one obtains a sequence of polynomials PN of degree k = 3 with parameters u = 4.945. * *

and

r = 1.426.. . ,

and this gives (63). The improvement in the bound for p(r) arises because of a better estimate for the common denominator of the rational coefficients in the representation of the integral as a sum of residues. Also in [1993a], Hata used a new construction of simultaneous approximations to r and In 2 to prove the inequality Ip+q7r+rln21

> H-p--E)

H = m=4lql, Id>,

/I = 7.016045.. . ,

110

Chapter

2. Effective Constructions

in Transcendental

which holds for any E > 0 and for all integers particular, this inequality implies that p(n)

5 8.016045..

Number

p, q, r such that

Theory

H 2 Ho(e). In

. .

4.6. Approximations to Algebraic Numbers. We shall give another applicain [Mahler 19311. Let ~1, . . . , rk tion of the identity (41), which first appeared complex numbers. In (41) be positive integers, and let WI,. . . , wk be distinct we set m = ri+.“+rk and ni = ... = n, = 0, and we take oi,...,o, to be the numbers wj + h for 1 5 j 5 Ic, where h runs through all integers 0 5 h < rj. If we further take z = ln(1 - z), 2 # 1, then (41) becomes the relation

E(z) =& s,0;gdc=

eQj(Z)(l

-

5)wj

,

(64

j=l

where

Q(C)

= nt=i

nziL’(<

-

wj

- h) and

Qj(z)

E (C[X], degQj

5 rj - 1.

Since E(z) = R(ln(1 - xc>), it follows that ord,=eE(z) = -1 + C,“=, rj, i.e., in the functional sense (64) gives good simultaneous approximations for the set wj = (j - l)/rC in this set of functions (1 - z)~‘, . . ., (1 - z)‘“~. Mahler identity, and obtained an analogue of (57) with the function (1 - z)jlk instead of In z. He used (64) to prove a special case of Siegel’s Theorem 1.14.

Theorem 2.12. Let t = (u/b) ‘In be an algebraic number of degreen, where a and b are natural numbers, let E be an arbitrary positive number, and let k be an arbitrary natural number. Then the inequality

has only finitely

many

rational

solutions

p/q.

In the proof of Theorem 2.12, the identity (64) replaced Siegel’s noneffective construction of an auxiliary polynomial in 2 and y. Note that Theorem 2.12 is non-effective, i.e., its proof does not enable one to find an upper bound for the numerators and denominators of the rational numbers p/q that satisfy the inequality in the theorem. This sort of weaker then Roth’s theorem 1.23 but effective result is very important in applications to Diophantine equations. In [1967a] Baker used the identity (64) with wj = mj/k, where the mj are distinct natural numbers in the interval 0 5 mj < k. Following Mahler, he proved an effective lower bound for a linear form in the numbers cj = (a/b)mj/k, 1 5 j 5 k, with integer coefficients (see $3.7 of Chapter 1). In the same paper he also obtained several other effective results. In [1979c] (see also [1983a]), Ch u d novsky made a detailed study of the asymptotic behavior of E(z) and the coefficients Qj(Z) in the identity (64) for fixed 2 and wj and increasing rj, where all the rj are of roughly the same

$4. Applications

of Hermite’s

Simultaneous

Functional

Approximations

111

size (e.g., rj = T +aj with fixed aj and increasing r). In the case of rational exponents wj , Chudnovsky was able to use Mahler’s explicit expression for Qj (z) to give a rather precise bound for the common denominator of the coefficients of Qj. These coefficients contain products of the form V(V + 1) a.. (V + N), where the v are rational numbers and N is an increasing integer parameter. For this reason the common denominator is bounded in terms of products of primes lying in various arithmetic progressions, and the asymptotic behavior of this bound can be determined using versions of the Prime Number Theorem for arithmetic progressions. The final result involves a positive term denoted (Chr); that depends on the common denominator n of WI,. . . , wk. For example,

c l
(d,n)=l

where cp(n) is the Euler p-function. In particular, 7r/2. Chudnovsky proved the following result.

(Chr):

= (7&)/6,

(Chr):

=

Theorem 2.13. Suppose that E is a positive number, n, s, k, a and b are positive integers, n > k, n > s, (n,s) = 1, (a, b) = 1. Let n be a common denominator for w1 = 0, w2 = s/n, . . ., wk = (k - l)s/n; and let (Chr): be the corresponding number (see above). F!urther suppose that

Then there exists an eflective constant qo depending on E, a, b, s, and n, such that for any rational number p/q with IqI 2 qo one has

/(;)“‘“-;I

2 ,Q,-x-E)

where (66) and

B=lfi-+%-I,

A=

max c: ck=l

In this theorem k is a parameter that gives the number of wj that appear in (64), and also the degree of the polynomials in Proposition 2.1. This number can be chosen as a function of a and b in such a way that the exponent x is minimal. In some applications where a and b are large compared to n, a suitable value of k can be determined from (65). In the case k = 2, note that Theorem 2.13 gives an improvement of the main result in [Baker 1964c]. It is because of a more precise computation of the common denominator of the coefficients of the polynomials Qj(z) that

Chapter

112

2.

Effective

Constructions

Theorem 2.13 has a better the concrete consequences

in Transcendental

Number

Theory

exponent x. The following inequalities are among of the results in (Chudnovsky 1979c, 1983a]:

fi - !! > q--2.429, s - !! 2 q--2.692, QI QI I I for all rational numbers p/q whose denominator tive bounds. (See also 53.6 in Chapter 1.)

is greater

55. Bounds for Rational Approximations of the Gauss Hypergeometric Function Functions

than

certain

effec-

of the Values and Related

Using the facts in $3.2, one can obtain Pad& approximations for e*, ln( l+z), and other functions related to the Gauss hypergeometric function. In this section we describe some number theoretic applications of these functional approximations. Here we shall not dwell on the results on approximating algebraic numbers that were already discussed in $4.6 and in Chapter 1.

5.1. Continued Fractions and the Values of ez. In $1 we mentioned the first proof the continued

of the irrationality fraction

of e was essentially

e = [2,1,2,1,1,4,1,1,6,1,.

. .] = [2,1,]11,>r

In [Davis 19781 this continued fraction precise bounds for rational approximations then: 1)

For any p, q E N with

There

exists

an infinite

sequence le-$I<

1, = (--ljn(qne for n 2 1 one has:

.

expansion was used to obtain very to e. Davis proved that if E > 0,

(i--e)%. of rational

numbers

p/q such that

(i+.c)z.

To prove his result, Davis used explicit integral numerators and denominators of the convergents p,/q, fraction, and also for the numbers

Namely,

-

that who found

q > qo(&) one has le--ii>

2)

due to Euler,

representations for the of the above continued

- pn) = /qne - p,l

.

$5. Approximations

of the Values of Hypergeometric

13~2

1 = -J

h--1

=

l J0

1

l

2

ettn(l

Functions

113

- t)ndt ,

ettn+l(l

- t)ndt ,

J 0

1 13n = n!

l ettn(l o

J

- t)n+ldt

(qn and p, are given by analogous formulas involving improper integrals). This type of integral had been studied as early as 1873 by Hermite in connection with his proof of the transcendence of e (see $2). Davis obtained similar inequalities for ]e2jt -p/ql, t E N (see [Davis 19791). In [1985] Popov extended Davis’ inequalities to imaginary quadratic fields. Let m E N, K = Q(e), b # 0, b E ZK. Popov proved the following assertions: 1)

For any E > 0 there exists qo(E) such that for anyp, q E ZK with one has

2)

For any E > 0 the inequality

has infinitely

many solutions

IqI 2 qo(.z)

p, q E Zn<.

In his proof Popov used numerical obtained using the integral

approximating

forms

for ellb that he

R > la1 > 0 . (In [1976a] Galochkin 11.1

had obtained

a weaker

result than Popov’s

assertion

5.2. Irrationality of z. We shall return to our discussion of rr (see $2) in order to describe another approach to studying its properties. The irrationality of x was first proved in 1766 by Lambert (see [Lambert 1946]), using the continued fraction for tanx:

tanx=II+-II+-II+... l/x 13/x 15/x

.

Lambert believed that, if x is rational, then this value must be irrational because the continued fraction is infinite. He concluded that if x # 0, then x and tans cannot both be rational. Since tan(n/4) = 1, this means that x is irrational. A rigorous justification for Lambert’s argument was given by Legendre [1860], who proved the following lemma.

114

Chapter 2. Effective Constructions

Lemma 2.3. Let mi,ni continued fraction

in Transcendental

E Z, mini

# 0, Irni/nil

m21 ml -c+-+-+*** In2 is infinite,

then it converges

1713 number.

Legendre also proved that 7r2 is irrational. sion

4

< 1, i = 1,2,. . . . If the

m3l

to an irrational

He did this by using the expan-

-221+-221+...

tanz7+

Number Theory

I3

,

(67)

I5

which is equivalent to Lambert’s continued fraction expansion. The relation (67) can be obtained from the continued fraction expansion of $J(c; .z)/$(c + 1; z) (see §3.2), since sinx = xr,!~(3/2; -x2/4) and cosx = $~(1/2; -x2/4). If we suppose that x2 = p/q, q > 0, is a rational number and substitute z = x in (67), we find that ()=-=

11 -PI -pql -pql i + 13q + I 5q + I 7q + ‘. . ’

tann

7r

contradicting Lemma 2.1. We give one more proof that 7~ is irrational. It is due to Hermite, technique is close to the methods in 52. Using integration by parts, n E N we have

and its for any

1

J

K2n+l

P(1 -

t)”

sin(rt)dt

= n! Q(r)

,

0

where Q(z) is a polynomial of degree n with integer coefficients. If 7~ = a/b, a,b E Z, b > 0, then b”Q(r) E Z and Q(r) # 0, since the integral is nonzero. We hence have the following inequality for some constant A: 1

1 5 b”l&(~)I

J

= (n!)-lb”rr2”+l

tn(l - t)nsin(7rt)dt

5 $

.

0

This is false for n sufficiently large; hence, 7r must be irrational. Although these two proofs at first appear to be completely different, they really have much in common. One can prove that for n 2 0 the numerators P,(x) and the denominators Qn(z) of the convergents to the continued fraction (67) satisfy the identity x2n+l

2n _ I

J l

77,. 0

P(1 - t)” cos((2t

If we now substitute

- 1)z)dt

= Qn(t) sinx - P,(x)

cosx .

x = ~12 in (68), we find that 1

n2n+l

1

n-1

J 0

tn(l -

t)”

sin(rt)dt

= 2”+‘Qn(7r/2)

,

(68)

$5. Approximations

of the Values of Hypergeometric

Functions

115

which is what Hermite used in his proof that r is irrational. It is easy to verify that Q,,(z) depends only on even powers of z. Hence, Qn(7r/2) = R(7r2/4), where R E Z[z], and this enables one to prove irrationality of x2 using Hermite’s argument (which is what he in fact did). 5.3. Maier’s Results. Maier’s long article [1927] was devoted to proving the irrationality and estimating the irrationality measure of various numbers. His basic tool was Pad6 approximation and closely related functional approximations. Maier proposed a new method for constructing such approximations. He proved, among other results, the irrationality of the numbers

and (A + 1,:1

+ 1)”

for any nonzero a E Q. In the last case we are using the notation (a), = a(a+l)...(a+v--1) and are assuming that X and p are rational numbers that are not negative integers. One of the results in [Maier 19271was that the numbers 1,

t1 (a>

and


(6%

are linearly independent over Q for any nonzero (I, p E Q. To prove this, Maier used the identity

and for any n E N gave an explicit construction of polynomials S(z), Hi(z) and HZ(Z) in Q[z] of degree at most 7n such that

i.e., he constructed functional approximations similar to Hermite-Pad6 approximations of the second kind. For large n the rational numbers Hi (1)/S(l) are good simultaneous rational approximations to (1 (o) and and H2(1)/S(l)
116

Chapter 2. Effective Constructions

at rational

points.

He proved

in Transcendental

Number Theory

that F(a, b,c; (Y), a, b,c,o

E Q, is irrational

under certain conditions; in particular, he showed that 1)

for any nonzero rational

number

(Y = p/q with

q > 2’e21p13 the value at CYof the dilogarithm L2(z)

=

5

5,

I4

5

1 ,

v=l

2)

is irrational; if a, b E Q, M (a + 1)” v=. (b + lb

F(a+l,1,b+1;z)=~~z

”

b # -1, -2,.

’

.. )

Q! = p/q E Q, (Y # 0, and q > p/2 + clp2, where cl is an explicit constant depending on a and b, then F(a + 1, 1, b + 1; o) is rational if and only if a-bENI). The constructions and results of Maier have been developed further in recent years (see $7.2). It was Apery’s proof of the irrationality of C(3) (see $5.6) that was the stimulus for this later work. 5.4. Further Applications of Pad6 Approximation. In [1979] Alladi and Robinson refined a result of Maier on values of the logarithmic function. To construct a Pade approximation for ln(1 - z) they used the Jacobi identity (see §3), which can be rewritten in the form Q,(i)ln(l

-z)

- iPn(i)

where Q,, and P, are polynomials is the Legendre polynomial

= zn/d’

~~(‘,~~,dt,

of degree n and n - 1, respectively,

&n(x) =-$(-$) nVP-XT> . In [Alladi, Theorem

Robinson

(70)

19791 the following

2.14. Let p, q E N satisfy

theorem

and Qn

(71)

is proved.

the inequality

4~-~)2 34.5) is bounded as follows:

and its irrationality

exponent (see

$5. Approximations

of the Values of Hypergeometric

Functions

117

The irrationality exponent was estimated using an analogue of Proposition 2.1 with Ic = 1 (see 54). In the same year as the article by Alladi and Robinson, the paper [Nikishin 1979b] appeared with a proof of effective bounds for linear forms in the logarithms of certain rational numbers. Nikishin used a refined version of the Jacobi identity to construct simultaneous rational approximations to different values of the logarithm. In the case of a single logarithm, his construction coincides with the one in [Alladi, Robinson 19791. Theorem 2.14 strengthens a result of Danilov (see (59)). Note that if one replaces z by ln(1 - z) in Danilov’s functional construction, one obtains Pad6 approximations for ln( 1 - z). A result equivalent to Theorem 2.14 was also proved in [Chudnovsky 1979bJ. Chudnovsky used the same functional approximations, but he carried out the entire argument by means of differential and recurrence equations that are satisfied by the sequences of remainders, numerators, and denominators of the Pad6 approximations. In our special case these recurrence relations have the form

(n + l)Qn+l(x) + (2n + lIPa: + 1)&n(x) + n&n-l(x) To find upper bounds for R, and Qn(-1), variant on Perron’s theorem.

= 0.

(72)

Chudnovsky used the following

Lemma 2.4. Suppose that the linear recurrence relation g ai(n)x,+i i=o

= 0

(73)

with variable coeficients ai has the property that each function ai approaches a constant ai as n + 00, and supposethat the limiting characteristic equation &xi i=o

= 0

has roots X1, AZ,. . .,A, with IAll > [AZ] > ... > 1X,1. Then there exist m linearly independent solutions z&f) of (73) such that t In I&f)I +

In IXjl,

j = l,...,m,

as n + co, and the solution x, of (73) satisfying

118

Chapter

2. Effective

Constructions

in Transcendental

Number

Theory

is unique up to a constant factor. It follows from Theorem 2.14 that all of the numbers In (1 + A), m = 1,2,. . .) are irrational, and their irrationality exponent approaches 2 as m + co. In particular, p(ln2) 5 4.622-e. . The last inequality strengthens the bounds of Baker (see (48)) and Chudnovsky [1979c]. If we set 2 equal to C = e2Xi/3 in (70) and choose the branch of the logarithm for which ln(l+C) = 7ri/3, we obtain a sequence of rational approximations to r/a. This is how Alladi and Robinson [1979] prove that X/A is irrational and that &r/h) < 8.309986.-a . In this connection see also [van der Poorten 19791. In [1982] Nikishin proved a quantitative refinement of Maier’s result on irrationality of the values of F(1, a+ 1, b+ 1; .z) (see $5.3). He used a construction of Pad4 functional approximations by means of orthogonal polynomials to derive a bound for the irrationality measure of the values of this function for a, b E Q, z E Q(i). We shall briefly describe this construction, following [Nikishin 19821. Let X(z) be a non-decreasing bounded function on the interval (a, b). The Markov function i(z) for the measure dX is defined by the relation

i(z) =

J

b dX(x) a z-x

.

Let Qn(x) be a system of orthogonal polynomials relative to X on the interval (a, b). If the function X(x) increases at infinitely many points, then the sequence Qn(x) is infinite, deg Q,, = n. If we determine a sequence of polynomials P,(z) by setting

P,(z) =

-Qdx)dx(x, ,Y Jab&n(z)z-x

then

QnPn+l-

Qn+lP,, = const # 0 ;

and in a neighborhood of 00 we have the expansion Q,Jz)i(z)

- p,(z)

= -&

+ *. * .

Thus, the polynomials P,(z) and Qn(z) give a PadB approximation for the set of functions 1, i(z). Markov’s theorem then says that the sequenceof rational

$5. Approximations

of the Values of Hypergeometric

functions Pn(z)/Qn( z ) converges to i(z) caseswhen all of the moments

Functions

119

uniformly in the region Cc\ [a, b]. In

b

ck

=

J

x”dX(x),

k20,

a

are rational numbers, one can prove results on Diophantine approximation of the values of X(z). Here is an example of how these general considerations are used in [Nikishin 19821.Suppose that the function X(z) on the interval (-1,1) is chosen so that dX(x) = ho(l - s)a(l+

z)&

)

where (Y and p are rational numbers greater than -1, and -1

1

(J

ho =

(1 - 2)“(1+

z)%

Then the Markov function i(z)

i(z) = -&

. >

-1

has the form W,B+1,a+P+2;&),

and the corresponding Qn(z) are the Jacobi polynomials qfy(x)

+ 2)- 0 z(-1P

= (1 - X)+(1

n ((1 - z)*+“(l

$ s

+ q+q

)

>

which are orthogonal on the interval [-1, l] with weight ho(l - z)“(l + 2)“. As a concrete example, Nikishin gives a bound for the irrationality measure of values of the function 3 ‘jk. ( ) In [1986] Huttner used Pad6 approximations for G(z) = F(a, b, c; z)/F(a, b+ 1, c + 1; z) (see $3.2) to prove irrationality of G(z) under certain conditions on a, b, c, z E Q and to estimate the irrationality exponent. In [1987] he used Pad6 approximations for the function F( 1, b, b + 1; z) to study approximations to the numbers F(1, l/k, 1+ l/L; EZ) for integers k 1 2 and rational x. He also proved a bound for the irrationality exponent of these numbers. In particular, using the representation -1 z - dt 2 0 1+t3 and the formula

J J

l/2

0

dt 1+t3

= F&1/3,4/3;

-x3)

= l(ln3+7r/&), 6

he found that p(ln 3 + x/&) < 11.040133.. .. The paper [Huttner 19871contains several such concrete results.

120

Chapter 2. Effective Constructions

In [Habsieger

in Transcendental

19911 linear recurrence

relations

Number Theory

are used to prove a bound on

the irrationality exponent of the values of F( 1,1/2, (a + 3)/2; X) under certain conditions

on the rational

numbers

x and (Y.

5.5. Refinement of the Integrals. In [1983c] Chudnovsky gave a twoparameter family of recurrence relations connecting hypergeometric functions with different parameters, and announced several results that improved upon earlier bounds, in particular: p(ln 2) 5 4.134400029.

j&r/&)

..,

5 5.792613804..

. .

In [1987] Rhin gives the following interpretation of the approximations obtained from Chudnovsky’s equations. Suppose we replace P(l - t)” by another polynomial H(t) in the numerator of the integrand in (70). Under certain conditions on H(t) that are given in [Rhin 19871, the integrals 1

J

H(t)

o (1 - zt)n+i

also give a sequence of rational HI(t)

approximations.

= (t(1 - t))[0.88nl(t2

where [ . ] is the greatest integer function, 4.26517429. . .; and the choice H2(t)

= (t(1 - t))[‘.‘“](6t2

(74)

dt In particular,

the choice

+ 2t - 1)2[0.06n] , leads to the inequality

p(ln 2) 5

- 5t + 1)210.05n]

leads to the bound p(ln 2) 5 4.134400029. . .. Rhin then gives a rather complicated polynomial H(t) that leads to the estimate p(ln 2) 5 4.0765. . .. Similar considerations enable him to improve the bound for the irrationality exponent of 7r/&. In the same year the papers [Rukhadze 19871 and [Dubitskas 19871 appeared, giving significantly better bounds. Rukhadze proved that p(ln2)

5 3.893 .

She used the integral (74) with H(t) = ((n-k)!)-l(tn(l-t)n))n-k, where k = [n/7], to construct rational approximations. It turned out that with this choice of H(t) she could make a much better estimate of the common denominator of the rational coefficients of the polynomials analogous to Qn(x) and Pn(x) in (70). In her paper she obtained the inequalities p(ln(4/3))

5 2.559,

p(ln(3/2))

5 2.911,

p(ln(5/3))

5 9.012 ,

and she pointed out that a similar construction in the case of a twodimensional integral (see s5.6) can be used to prove that p(r”)

5 7.552 .

$5. Approximations

of the Values of Hypergeometric

Functions

121

In [Dubitskas 19871the bound ,+r/~)

5 5.516

is proved. To construct rational approximations to this number, Dubitskas used integrals of the form

Jl H(t)& I-J l-t+P

’

where H(t) = (k!)-‘(P(1 - t)n)(“) withn=5pandk=4pork=4p-2forp a prime of the form 6m + 1. In [1993] Dubitskas improved the above estimate, obtaining p(r/&) 5 5.2. These results are discussed in detail in [Hata 1990b], where the author uses the polynomials

(n+m-m’) H(t)= (n+,‘- m,)!(t-71 - v+m> 7 where m and m’ depend on n in some specified way, to prove several results on the irrationality exponents of various numbers. In particular, taking m = ,m’ = [n/7], he improves upon estimates of Rukhadze and Dubitskas: p(ln 2) 5 3.8913997. * . ,

&r/A)

5 5.0874625-e. .

In the same paper Hata usesmultiple integrals (see $5.6) to obtain the bounds p(r2) 5 7.5252-e-,

p(C(3)) 5 8.8302837. . . .

A consequence of the first inequality is that p(n) 5 15.0504. . .. In [Rhin, Viola 19931the authors prove that p(r2) 5 7.398537 -. - , and hence that p(w) 5 14.797074.. .. We also mention the bound p(n/a) 5 4.601579.. . obtained in [Hata 1993133.In the same paper Hata proves that for any E > 0 and p, q, r E Z with H 2 HO(E) one has Ip + qr + T In 21 2 Hmcve,

[ = 7.016045. +. .

In particular, it follows from this inequality that p(r) I: 8.016045. . .. 5.6. Irrationality of the Values of the Zeta-Function and Bounds on the Irrationality Exponent. It is well known that the value of the Riemann zetafunction C(s) = 2

v-8

v=l

at an even positive integer can be expressed in terms of K. Namely,

122

Chapter 2. Effective Constructions

in Transcendental

C(2k) = (-1)k-1$&B2t where BZk is a rational number. recursively from the relation

The Bernoulli

kr:‘)Br=O, r=O

Number Theory

, numbers Bz~ can be determined

n=

1,2,...

,

and the condition that Bo = 1. For example, C(2) = 7r2/6 and C(4) = 7r4/90. Since rr is transcendental, it follows that <(21c) is transcendental for k 2 1. It is then of great interest to determine the arithmetic nature of the values of the zeta-function at odd positive integers. It is conjectured that 1) 2)

the numbers 1, C(2), C(3), . . . , C(n) are linearly independent n 2 2; and all of the numbers <(2k + 1) are transcendental.

In 1978 Apery was able to prove that C(3) is irrational for the irrationality exponent of this number. Theorem

2.15.

C(3) is irrational,

over Q for any

and find a bound

and

p(c‘(3)) 5 13.4178202... . Apery published only a brief sketch of the proof of this theorem (see [Apery 19791). A detailed proof can be found in [van der Poorten 19791and [Cohen 19781. Consider two sequencesa, and b, that satisfy the recurrence relation (n + 1)3u,+i - (34n3 + 51n2 + 27n + 5)21, + n3u,-i

= 0

(75)

and the initial conditions a0 = 0, al = 6, b. = 1, bi = 5 , Surprisingly, it turns out that all of the b, are integers, and the rational numbers a, have denominators that are not very large. More precisely, if d, denotes the least common multiple of the first n natural numbers, then 2df,a, E Z . A well-known consequenceof the Prime Number Theorem is that d, 5 e(l+E)n for any E > 0. One can write out explicit expressions for the a, and b, from which one can deduce the above properties, as well as the fact that %/bl + C(3). Using (75) and the initial conditions, one can show that a,&-1

- an-lbn = 6nm3,

n21,

55. Approximations

of the Values of Hypergeometric

Functions

123

and

0 < C(3) - 2

=

g 6 = O(n-lb,l) k=n+l k3bkbk-l

.

Moreover, these relations imply that the numbers in the sequence U(3)

- an

are all nonzero and approach zero as n -+ 00. The equation (75) can be rewritten in the form (1 + n-1)3 u,+l - (34 + 51n-l + 27n-’ + 57~~~)~~+ u,-1 = 0, so that the limiting characteristic equation (see Lemma 2.4) has the form x2 - 34x + 1 = 0 . The roots of this quadratic equation are a = (1+

It follows from Lemma

and

&)4

p=@--1)4.

2.4 that

13, = 0 ((1 + ,,,)

and

b&(3) - a, = 0 ((h

- l)‘,>

.

(76)

But since 0 < &lb,5(3)

- a,[ = 0 (e3(l+qd2

- 1)4,>

)

E is arbitrary, and e(fi - 1)4 < 1, we can conclude that c(3) is irrational. addition, (76) along with Proposition 2.1 (with k = 1) implies that lncr+3 cl(C(3)) I 1+ ln(u-3

In

= 13.4178202... .

The most difficult point in this argument is the proof that all of the numbers b, and dia, are integers. To show this, one works directly with explicit expressions for a, and b,. One has

The proof of this equality, i.e., the verification that the sums on the right satisfy (75), is a little difficult. The expressions for a, are more complicated, and it is considerably more difficult to prove that those expressions satisfy (75). Proceeding in an analogous way, Apery proved irrationality of c(2) = 7r2/6. In this case the recurrence relation has the form (n + 1)2~,+1

- (lln2

+ lln

+ 3)u,

- (n - 1)2~,-1

= 0.

124

Chapter 2. Effective Constructions

in Transcendental

The sequences a; and bk are determined

initial

recursively

Number Theory

by this relation

and the

conditions ah = 0, ai = 5, bb = 1, bi = 3 .

Explicit

expressions

can be given for these numbers, b:,=f:(;)2(n;k),

e.g.,

n>l.

k=O

Using these expressions, one can prove that bL,diak proof is just as before. One finds that p(r2)

5 11.850782+..

E Z. The rest of the

.

In exactly the same way one can obtain a bound for the irrationality sure of In 2 = Li(-1) (see [van der Poorten 19791): p(ln 2) 5 4.622100832.. The recurrence relations fraction expansions:

that ApQy

mea-

. .

found give us the following

continued

n6 I 61 11 64 I C(3) = 5 - 1117 - 1535 - - *. - ]34n3 + 51n2 + 27n + 5 - ” ’ ’

n4 ]lln2

+ lln

I + 3 + ... *

In [1979] Beukers gave a new interpretation of Apery’s rational approximations to c(3). This led to what is currently the simplest proof of Theorem 2.15 - a proof in the spirit of the ideas in 85.3. We first show how the irrationality of C(2) is proved in [Beukers 19791. One has 1

IS

’ (1 - Y>“&&) dxdy = a; - bLc(2) , (77) 1 - xy 0 0 where Qn(x) is the Legendre polynomial (71) and a; and bk are the same as in Apery’s proof that C(2) is irrational. This equality is not essential for the proof; the only important circumstance is that we have relatively simple expressions for a; and b; that can be used to show that bk, d2,ak E Z. We have 0 < dilak - b;C@)l < &((& - 1)/2)5”<(2) < (5/6)” , (78)

from which it follows that C(2) = 7r2/6 is irrational. The proof in [Beukers 19791 of the irrationality of c(3) is analogous. the relation

Qn(x)Qn(~) dx & = 2(an - M(3))

Prom

(79)

55. Approximations (with

of the Values of Hypergeometric

the same a, and b, as in Apery’s 0 < 2dila,

- b43)I

proof)

Functions

125

we obtain:

< 2<(3)d;(h

- 1)4n < (4/5)”

These inequalities imply that c(3) is irrational. One can obtain upper and lower bounds for the integrals because, like (70), they can be transformed to the form

. (77) and (79),

II010’zn(l -z)nyn(l -?d”dzdy (1 - .y)n+’

and

JJJ

01 0 1 0 l P(l

- (1 z)nyn(l- (1 - zy)ur)n+’ y)%P(l-

respectively. This is the most cumbersome part of the proof. In $5.5 we mentioned that, by modifying these multiple [199Oa] was able to prove that p(n”)

5 7.5252..

-,

p(c(3))

7

“)“dzdydW

integrals,

Hata

5 8.8302837. . . .

We now describe one more variant on the argument that gave Apery’s sequence of rational approximations to c(3). The following theorem was proved in [Gutnik 19831. Theorem

For any nonzero q E Q at least one of the numbers

2.16.

d(2) - 3<(3),

C(2) - 2q In 2

is irrational. An important role in the proof is played by the polylogarithms

and by the so-called Meijer functions (see [Luke 1969]), especially + 1 + s) Z8dS F(z)= & Jc F4(-s)F2(n T2(n + 1 - s) =-

1 2ni

(80)

:p$l-.~;s”~)) 2C&J2 ZSdS 7 SC c

where the contour C begins and ends at -00, enclosesthe negative integers in the positive direction, and does not contain any of the nonnegative integers. If we compute this integral by adding the residues of the integrand at -1, -2,. . ., we obtain the following formula for ]z] > 1: F(z) = G(z) + (ln z)H(z)

,

(81)

126 where

Chapter 2. Effective Constructions

G(z) = 2A,(z)L3(z-7

in Transcendental

+ B,,(z)L~(z-~)

Number Theory

+ G(z)

H(z) = A,(z)L2(z-l) + &(z)L~(z-‘) + h(z) and A,, B,, C,, D, E Q[z] have degree at most n. Here H(z) = -g R(k)z-k,

G(z) = 5

k=l

R’(k)z-’

, ,

(82)

,

k=l

where

R(t) =

(t - 1) * * * (t-n) 2 t(t + 1). . . (t + n) > ’

so that ord,=,H(z) 2 n + 1, ord,=,G(z) 2 n + 1. This means that the polynomials A,, B,, C,, D, give simultaneous functional approximations in a neighborhood of co for the functions 1, Li (z-l), Lz(z-l), L3(ze1). There are explicit expressions for these polynomials, and in particular one has An(l) = b,, Cn(l) = -2an, and B,(l) = 0. Thus, letting z + 1, from (80)-(82) we obtain

(83) 2(M(3) - an) . Incidentally, this equality can also be proved directly, by computing 1, as a sum of residues. But we wanted to show the connection between this construction and functional approximations. Once we have this formula, we can prove irrationality of c(3) as follows. Using the saddle-point method to determine the asymptotic behavior of 1, as n + +oo, we find that

=

b&(3)

- a, - cn-3’2(d3

- 1)4n )

where c is a positive constant. If we now use the explicit expressions for polynomials A,(z) and &(z), we can easily prove that b, E Z and obtain a bound on the denominator of a,. As we saw earlier, this leads to irrationality of c(3) and to ApQry’s bound for the irrationality exponent. It would be curious to find an elementary proof that C(3) is irrational the spirit of Fourier” (see §I), based on the relation

2

the also the “in

R’(k) = 2(b,c(3) - a,) .

k=l

It would suffice to obtain a “good” bound for the left side and prove that the left side is nonzero. In [1981] Beukers used similar functional approximations to prove the irrationality of c( 3)) and showed that the approximations are uniquely determined up to a constant factor.

$6. Generalized Hypergeometric

56. Generalized

Hypergeometric

127

Functions

Functions

It is natural to try to find generalizations to other classes of functions of Hermite’s identities (41) and (40), which in the case of the exponential function were used to obtain Hermite-Pade approximations of the first and second kind, respectively. Such a generalization was found for a class of functions that includes the Gauss hypergeometric function. Definition 2.1. The generalized hypergeometric a1 )...) ap,bl )... , b, E C is given by the series P F9

where

the symbol

al,...,ap ( h,...,bq

>

= 2 (al), n=O Ml

. . . (a,), *** @&I

with

parameters

$ 72. ’

(84)

((Y)~ is defined by setting (a),

(Q)O = 1, Here one assumes

;z

function

= cx(a + 1). a. (c~ + n - l),

n = 1,2,. . . .

that bj # 0, -1, -2, . . ..

If p 5 q, then the series (84) gives an entire function. This is the case that will be considered in this section. When p = q + 1 and aj # 0, -1, -2,. . ., the series (84) has a finite radius of convergence; that case will be examined in §7. The function in (84) can also be represented in the form of an integral in the complex plane: PF 9

al,...,ap ( h,...,bq

;z

=

>

r(h) * - * T(bq) r(al) . . . r(a,) . T(al + s) . . . T(a, + 8) .- 1 T(-s)(-z)sds, 27Ti s L T(bl + s) * . . qb, + s)

where r(t) is the Euler gamma-function. Here the contour L begins and ends at +oo, goes once around each of the poles of F(-s) (i.e., the points 0, 1,2,. . .) in the negative direction, and does not contain any of the poles of r(aj + s). If p 5 q, the integral converges for any z # 0; and if p = q + 1 it converges for 0 < ]z] < 1. In the latter case we may also take L to be a contour that goes from 0 - ioc to (T + ioo and separates the poles of r(-s) and r(aj + s); and one can also give a multiple real integral representation for pFq (see $7). All of these ways of representing pFq have been used to explicitly construct functional approximations for these functions. The function (84) is a solution of a linear differential equation of order q + 1 with coefficients in C(z). The properties of the generalized hypergeometric functions are described in detail in the book [Luke 19691. Some of them will also be given in Chapter 5.

128

Chapter 2. Effective Constructions

6.1. Generalized

Hermite

Identities.

in Transcendental

Number Theory

We consider the function

where b(z) is a polynomial of degree m 2 1 with leading coefficient 1 that does not vanish at any of the positive integers. This function is a solution of the linear differential equation @(4 - ZJYY = w

7

(85)

where E is the identity operator and 6 is the operator 6 = zg. (z + Ai). . . (z + A,), then f(z) can also be written in the form l

f(4=lqXlfl 7”‘)

x

If b(z) =

+l;z). m

In [1966] Osgood proved some bounds for linear forms in the values of f(z) and its derivatives at distinct nonzero points wi, . . . , wt in the ring of integers Zr of a quadratic imaginary field I[. He assumed that all of the roots of the polynomial b(z) are rational numbers, i.e., that f(z) is an E-function (see Chapter 5), and that none of the points wj are conjugate to one another. The coefficients of the linear forms were assumed to lie in Zr. Soon after (in 1967), Shidlovskii proved just as strong a result for an arbitrary set of E-functions that make up a solution to a system of linear differential equations with coefficients in Cc(z) (see Theorem 5.26). Nevertheless, Osgood’s paper [1966] turned out to be very useful for later work, because it gave explicit constructions for the approximating linear forms. These constructions generalized the earlier ones of Mahler, who had been working in an analogous situation with the function eZ. Let p be either 0 or 1, and define g(z) as follows: g(z) = zP(z - WI)*. . . (z - wtp

.

For every N E N we determine n, h E Z so that

N = (mt + p)n - h,

n>l,

O
zV(4 dz pN=Lsc-(g(z)>” ’

We then set

27n

where C is a smooth closed contour that goes once around each zero of g(z). If we express this integral as a sum of residues at 0, WI,. . . , wt, and use the fact that f(z) is a solution of the linear differential equation (85), we can show that t

m-l

PN

=

a +

c

c

r=O

j=l

&jf"'(Wj),

a,a,j

E I[ .

636)

56. Generalized Hypergeometric Functions

129

Using Osgood’s explicit construction, one proves that each of the numbers f(‘)(~j), 0 5 r < m, 1 5 j 5 t, is a linear combination of the linear forms (86) with indices N, N + 1,. . . , N + (m + 2)(mt + p), and in the case p # 0 or g(0) # 0 such a representation exists for 1; hence, those linear forms are a complete set of linearly independent forms. After finding an upper bound (depending on N) for the denominators of the coefficients a, arj and also for the absolute values of the coefficients and for ]PN], one can obtain a lower bound (depending on the absolute values of the coefficients) for an arbitrary linear form in the numbers f(‘)(wj), 0 5 r < m, 1 5 j 5 t, in much the same way as in $1.4. In [197Oa] Galochkin obtained bounds for linear forms in the values of the functions

= g

(Al + 1)“. .‘. (Am + 1) ” C>mv

and = CP-lv(Z),

Cpj(Z)

j = 1,2 ,...,

m,

(87)

where 6 = .z-& as before. Theorem 2.17. Let XI,. . . , A, be rational numbers not equal to -1, -2,. . ., and let (IY be a nonzero algebraic number such that a” E Il. Then for any bj E Zn, 0 5 j 5 m, SU& that max lbjl=H>3 --

O<j<m

one has Ibo + blcpl(a) + ... + bmpm(a)l > Cs~-m-7zm2(lnlnf4-’

.

This result strengthens Osgood’s bound. We note that the exponent in the bound for a linear form in the values of E-functions that can be proved by the Siegel-Shidlovskii method (see Theorem 5.27) is somewhat worse than in Galochkin’s theorem. In [Galochkin 197Oa] the functional linear forms are constructed explicitly. Let E be the identity operator, n E N, M = (m + l)n, and M-l+j Rj(Z)

=

n

k=n+l

(6

-

1

mkE) v=n+l

(xl

+l)Vo*'(Xm

+

1)~

for j = O,l,..., m. It can be verified that the Rj(z) are linear forms in Lcpl(Z),.. . , pm(z) that are linearly independent over C(z) and have polynomial coefficients Pji(z) of degree at most mn. In addition, each Rj(z) has a zero at z = 0 of multiplicity at least (m + l)mn, and the coefficients of the

130

Chapter 2. Effective Constructions

in Transcendental

Number Theory

polynomials Pji(z) satisfy all of the arithmetic properties that are needed for the estimates. In [Galochkin 19791 Theorem 2.17 is generalized to linear forms in the values of (p(z) and its successive derivatives at pairwise distinct nonzero points WI,... , wt E I[. The power of H in the estimate again differs by 0 ((In In H)-l) from the correct exponent (by which we mean the negative of one less than the number of terms in the linear form). Galochkin uses a construction of the auxiliary functional forms that on the surface seems different from the one in [Galochkin 197Oa]. Namely, let n E N, and let ~1,. . . , rt E Z with 1 5 rj 5 m. Consider the function cpbW6

R(z)

27Fi

=

J-

r

p+l(<

-

wl)mn+‘l

.

.

.

(6

-

Wt)mn+rt

’

f

where the integral is taken around a circle r centered at 0 and containing all of the points WI,. . . , wt. Note that when z = 1 this integral is close to the integrals in [Osgood 19661. The function R(z) has the following properties: 1)

its order of zero at 0 is equal to N = n(mt + 1) + ri + .-. + rt ;

2)

R(z)

= PO(Z) + C:=,

PO(~),

pkj(z)

E c[z],

Cj”=,

Pkj(z)cpj(wkz),

where

deg PO(Z) = n,

deg

pk j (

3)

]R(l)]

pj(z>

= @iv(z),

and

In =n
ifjrk;

< erNNemN.

Galochkin constructs a complete system of linearly independent functional linear forms by varying the rj in the exponents in the integral representation for R(z). (This is the same technique that Mahler used; see 54.) Galochkin then uses the arguments in $1.4. The constructions of Hermite-Pade approximations in [Galochkin 197Oa] and in [Galochkin 19791 actually have much in common. If no, ni E No, then any entire function G(z) = &,z” v=o satisfies the following coefficients:

identity,

as is easily verified

by equating

ni!

k;fp - w (“ZOWV) = -:27r2 f r 0 where r is a closed contour

containing

0 and 1.

p+l(c

power

G(zCPC - 1)n1+1 ’

series

$6. Generalized Hypergeometric Functions

131

6.2. Unimprovable Estimates. Can one improve the exponent in the bounds for linear forms by replacing 0 ((lnlnH)-‘) by a more rapidly decreasing function? Results from the metric theory of Diophantine approximation (see [Sprindzhuk 19771) show that for almost all e = (01,. . . ,0,) E R” in the senseof Lebesgue measure and for all H sufficiently large, one has lblf& + . . . + b,8,1

> H1-n(lnH)-m-E,

H=max]bi],

for any E > 0 and for all H 2 Ho = Ho@, E). In [1976a] Galochkin was able to prove bounds with this exponent for certain values of the functions pj(z) (see (87)). Theorem 2.18. Suppose that m 1 2, the algebraic numbers Xi,. . . , A, are not equal to -1, -2,. . . and are such that P(z) = x(z + Xl) *. . (z + X,-l)

9 = m4gll,.

. . , lib-11),

= P + gm-lP-l

+ * * * + g1z E Jl[Z] )

and b E Za, b # 0. Then for any hj E Zp, 1 5 j 5

m, such that mm

l<j<m --

IhjlIH,

HZ3,

one has the inequality Ihlcpl(l/b)

+ ... + hmcPm(l/b)I > cll(lblH)l-”

(88)

where 75 = (m - 1)2g - (m - l)Reg,-i + m(m2 + m - 4)/2 and cl1 e’sa positive constant depending only on m and g. When b, gi, hi E Z, the bound in Theorem 2.18 is, except for the exponent 75, the same as in the result for almost all numbers. The proof of Theorem 2.18 usesthe same construction of functional linear forms as in Theorem 2.17 with A, = 0. The forms one uses are Rj(Z) =

(mn-l+j)! 27ri

cpM)dC‘ r (C 1)mn-1+j ’ f

j = l,...,m,

where r is the circle ]C - I] = 1 taken in the counterclockwise direction. Is it possible in certain cases to further improve (88), i.e., to get a better ys? In [1986] Korobov used recurrence relations to prove precise lower bounds for linear forms in the values of certain generalized hypergeometric functions at the point l/c, c E Z, c # 0. These estimates differ only by a constant factor from the upper bounds that are possible for these linear forms. Note that Korobov’s theorem was proved in 1981 and published only in 1986. In [1984] Galochkin used an effective analytic construction of approximating forms to generalize Korobov’s results and prove unimprovable bounds for the values of the functions p(z) in Theorem 2.17. We now give an exact formulation of these results.

132

Chapter 2. Effective Constructions

in Transcendental

Number Theory

Suppose that s E Z, s 2 2, Xi,. . . , A, are elements of the quadratic imaginary field II that are not equal to -1, -2, . . ., and a is a nonzero number such that a, aXi, . . . , aX, all belong to Zn. Let Z”

ti’(z) = g

v=o arn”y!(X1

where m = s + 1. Set uj = {ReXj}, a real number, and set j-1 -

A = min l<j<s --

+ 1)“. . . (X, + l)V ’

where

{ . } denotes the fractional

+ uj _ Cl + * *. + us S

s

part of

. >

In Theorem 2.19 there are also some positive numbers pl ,p2, n that depend on Xi,... , A, and are computed in [Galochkin 19841 in a way similar to A. For brevity we shall not give the explicit expressions for these numbers. Theofern 2.19. Suppose that b E 251,b # 0, ho,. . . , h, E 251are not all zero, max Ihi 1= H > 3, and

R = hs$(l/b)

+ h&(1/b)

+ ... + h,@)(l/b)

.

‘Then there exist effectively computable positive constants cl, ~2, ~3,q depending 012X1,..., X,, a and b, such that 1)

one has

2)

there exist infinitely many forms R for which

]RI > ci HmS(ln H)-‘(‘-“)

(ln In H)pl ;

IRI < c2HmS(ln H)-“(l-A)(lnln 3)

if

H)pl ;

R is a primitive form (see [Galochkin 19841for the definition) and if IRI > csH-‘(In

H)-S(‘-A)(Inln

then IRI > c4H-‘(In

H)P2 ,

H) rl- 41-A)

In comparison with [Galochkin 197Oa],the sequenceof approximating linear forms in the proof of Theorem 2.19 is explicitly constructed so as to be s times denser. For every N E N define T, t E Z by setting l
N=st-r+l,

The approximating forms needed to prove Theorem 2.19 are given as follows: RN(Z) = at fi(6

- kE)$,(z)

k=l

= &+P’ t’

$44M

2ni f r C(C - l)t+l

’

$6. Generalized Hypergeometric

Punctions

133

where @i(z) = C&$(Z), $j(z) = aj-‘(6 + XiE)...(6 + Xj-iE)$i(z) for j = 2 >‘.‘, s, and r is a contour that goes once around 0 and 1. These forms are linear combinations of the functions @c(z), +i (z), . . . , $*(z) with polynomial coefficients, and any m = s + 1 successive forms are linearly independent over C(z). It is from these forms at the point l/b that one obtains the required numerical approximating forms, which lead to a proof of part 1) of Theorem 2.19 and to an infinite sequence of forms satisfying the inequality in part 2). In [1990] Korobov used the Jacobi-Perron algorithm to find multidimensional continued fraction expansions for certain vectors connected to the values of generalized hypergeometric functions. These were multidimensional generalizations of certain classical continued fractions of Hurwitz, Euler, and Ramanujan. As a consequence Korobov was able to prove some unimprovable bounds for linear forms. Here is one of his results: Let s, q E N and set ‘$‘k

=g

v=.

((s

+

1)v

+Ik)!g(s+‘)y+”

k=0,1,...,

’

s.

Then there exists a positive constant cl such that the following inequality holds for any integer H > 3 and any ho, . . . , h, E Z not all zero, where maxihi] 5 H: --s(s+3)/(2s+2) Iho$o + . . . + h,&l

> c~H-~

In addition, there exists an infinite sequence of linear forms in the same numbers for which the opposite inequality holds with a different constant cq. In particular, for s = 3 one has the precise bound ]ho cosh(l/q)

+ hl cos(l/q) + h2 sinh(l/q)+hs

sin(l/q)]

>

> c~H-~

6.3. Ivaukov’s Construction. Generalizing Galochkin’s results, Ivankov [1986, 1991b] carried out a similar effective construction of simultaneous approximations for the values of arbitrary hypergeometric E-functions. Let cyi, . . . , a,. and pi,. . . , &, be rational numbers not equal to -1, -2,. . ., 0 5 T < m, and let F(z) be the generalized hypergeometric function with parameters 1, cri + 1,. . . , crp + 1 and pi + 1,. . . , pm + 1. If we denote a(x) = (57+ al) * *. (z + a,)

and

we have F(z)=gzvk@lg. v=o (The term for v = 0 is taken to be 1.)

b(z)

=

(2

+

Pl)

. . . (x

+

Pm)

,

134

Chapter

2. Effective

Constructions

in Transcendental

Number

Theory

Theorem 2.20. Suppose that none of the di$erences cxi - pj are rational integers, and WI,. . . , wt are pairwise distinct and nonzero elements of I. If bo, bij, 1 5 i 5 t, 1 5 j 5 m, are any nontrivial set of integers of II having absolute value at most H, H 2 3, then

b. + 2 2 bkjF(j-l) (wk) > ~10H-mt-Y4(~~~~~)-’ , kc1

j=l

where 74 is a constant depending only on the (Yi, bj, and wk. In the case T = 0 this result was proved in [Galochkin 19781. We now describe Ivankov’s construction in more detail. The functional approximating form R(z) needed to prove Theorem 2.20 is defined as follows:

(89) where r = mt(n + 1) + n,

k=l

in which the Bk(V) are polynomials in v of degree at most (n + 1)m - 1. In addition, R(z)

where PO(Z), the notation Xl@:>

=

Pkj

=

PO(z)

+

2

2

k=l

j=l

pkj(z)Fj(wkz)

,

(91)

(z) are polynomials in z of degree at most n. If we introduce

Xj(z)

1,

=

Cz + Pj-l>Xj-1

Cz>

for

j =2,...,m,

then the functions Fj(z) in (91) are given by the formulas Fj(z)

= Xj(G)F(z)

= eZyXj-l(u) v=o

k@l $$

.

(92)

It turns out that it is more convenient to construct functional linear forms in the Fj(z) than in the PlF(z). In order to explain where (91) comes from, we consider the following m(n+ 1) polynomials fj,(V) in the variable V: n-1 fjp(v)

=

xj(v-p)

n k=p

P--l 4~sk)

n k=O

b(v--k),

llj<m,

O
(93)

$6. Generalized

Hypergeometric

Functions

135

These polynomials have degree at most m(n + 1) - 1, and one easily verifies that they are linearly independent over Cc.Thus, for some wkjp not depending on v one has the identities Bk(V)

=

9 p=O

l
~wkjpfjp(~)~ j=l

Prom (90) and (93) we now find that

In view of (89) and (92), we then obtain the representation (91) that we need with polynomials n Pkj(z) = c wkjpw~z” . p=o

Moreover, it can be shown that the coefficients of the polynomials PO(Z), Pkj (z) have a common denominator that is bounded from above by ecn, and the absolute values of these coefficients are at most ecnn(m--r)n. We also have the bound IR(l)j < n -tm(m--r)necn. As in the Siegel-Shidlovskii method (see Chapter 5), one uses differentiation to greatly increase the number of functional linear forms. The rest of the proof proceeds according to the usual scheme. It turns out that the sets of functions F(j-l)(z) and Fj(z), 1 5 j 5 m, are connected by linear relations with coefficients in C(z), and this enables one to express the linear form in Theorem 2.20 in terms of the Fj(wk). In the case t = 1 and wi = 1 the function @(v) in (90) is easy to compute:

If we make appropriate choices of the oi and /3j and equate coefficients of powers of z in the Taylor series, we obtain another representation for the function (89) (in the case when t = 1, wi = 1, and r = mn + m + n) as a generalized hypergeometric function: R(z) =zr(mn)!

zy

$i;. k

'r+l

Fm

1

(~1+ 1 + mn + m, . . . , a,+l+mn+m,mn+m+l /?I + (m + l)(n + l), . . . ,& + (m + l)(n + 1)

;z

. >

We note that in the case t = 1 this type of construction of Hermite-Pad6 approximations of the first kind was suggested in 1980 by Chudnovsky, who

136

Chapter 2. Effective Constructions

used the analytic

theory

in Transcendental

of differential

equations

Number Theory

(see [Chudnovsky

1980b,

198Oc, 1980f]). In certain cases these constructions also lead to quantitative results for functions with irrational parameters. For example, in the case of the function p(z) in Theorem 2.17 it suffices to require that the polynomial b(x) = (x + Xl) . . . (x + A,) belong to the ring ll[z]. For this type of result see, for example, [Galochkin 197Oa]; and for more general results see Ivankov’s papers [1993-19951.

$7. Generalized

Hypergeometric Series of Convergence

with

Finite

Radius

When p = q + 1 in (84), i.e., when the generalized hypergeometric series has finite radius of convergence, one can use the following representation of the function as a multiple integral:

P

FQ

4

a1,...,$I II j=l

bl,...,bg

lYbj) r(aj)r(bj

-

1

%)

*

l n;=‘=, t;‘-l(l - ti)b’+‘dti.. (1 - ti . . . tgZ)+ 0

.I J ...

0

.dt

(94)

7.1. Functional Approximations of the First Kind. In [1972] Fel’dman tained a lower bound for linear forms in values of the functions

f,(z) $& for different

parameters

P.

ob-

= $PJJ+~;~)

/.L.

Theorem 2.21. Suppose that ~1,. . . , pm are rational numbers that are not equal to 0, -1, -2,. . . and are not congruent to one another modulo 1, E > 0, and II is a quadratic imaginary field. There exists a constant y 2 1 depending only on PI,... , pm, E such that if a, b E & satisfy the condition 0 < e/lalm+l

< Ibl’=/(“+l+“=)

,

then one has

1x0+ afm (a/b) + ..+

+x,,J,,,(a/b)l

> 1x1 --.x,I-~-’

for any x0, x1, . . . , x, E ZH such that 1x1 . - . x, I 2 X0, where Xo is an effective constant depending only on ~1,. . . , pm,&, a and b.

$7. Hypergeometric

Series with Finite Radius of Convergence

137

The condition that a/b be close to zero is typical of almost all results on the values of functions whose power series have finite radius of convergence. It is an interesting but very difficult problem to study the values at points that are far from zero, for example, on the boundary of the disc of convergence. In this connection we note Apery’s theorem that Ls(1) = c(3) is irrational (see §5.6), and also Wolfart’s theorem on transcendence of the values of the Gauss hypergeometric function at algebraic points (see 54.4 of Chapter 3). If p = p/q is a rational number, then it is easy to verify that P-1

c = e2dq ,

fcl(z) = -zplq C Cwptln(1 - Ctzllq), t=o

so that in this case the numbers f,(a/b) in the linear form in Theorem 2.21 are linear combinations of the logarithms of certain algebraic numbers. The proof of this theorem is based on an explicit construction for any ENofpolynomialsP~(z),k=0,1,..., m, nl,...,n, de&(z)

5

nk,

n0

=

lFjF-nj --

,

that give sufficiently good functional approximations of the first kind for the set of functions 1, fpk (z-l), 1 5 Ic 5 m, at the point z = 00. The bound in Theorem 2.21 depends on each coefficient zk because of the use of functional approximations for any choice of natural numbers ni , . . . , n,. For more details on this type of bound, see 56 of Chapter 5. Theorem 2.21 was generalized in [1985] by Sorokin, who proved a similar result for the values of the functions F(1, pi, pi + ~0; z), 1 5 i 5 m, with positive rational ~0, . . . , pL,, pi $ pj (mod l), 1 5 i < j 5 m (see Theorem 2.23 below). Sorokin’s proof was based on an effective construction of Pade approximations of the second kind for these functions. In [1979a] Nikishin proved the following result on the values of the polylogarithms L,.(z) (see 55.6) at rational points close to zero. A similar but less precise result was a consequence of Galochkin’s general theorem on the values of G-functions (see 57 of Chapter 5). Theorem

2.22.

If 5 = b/a < 0 is a rational number and

bm+’ < ]a]exp(-(

m - l)(mlnm

+ 2mln2)) ,

then the numbers 1,

h(t),

J52(E),...,

Ln(O

(95)

are linearly independent over Q. This theorem generalizes Maier’s result on irrationality

of &(p/q).

The proof of Theorem 2.22 is based on a construction of Hermite-Pade approximations for the functions 1, &(1/z), &(1/z), . . ., &(1/z) in a neighborhood of 00. For n E Z, q E Z, 0 5 q < m, let R(t) be the rational function

138

Chapter

2. Effective

Constructions

in Transcendental

Number

Theory

t(t - 1) . . . (t - nm - q + 2)

R(t) = (t + 1)” ’ ’ * (t + n)“(t and let H,,,(z)

be given by the integral

where the contour C is the line Re t = -l/2 This integral converges for ]z] > 1. Then: 1) for certain polynomials Am, Kz,&)

=

+ 12+ 1p ’

Ao(z)

+

k = 0, 1, . . . , m, one has

A~(z)Ll(l/z)

2)

degAo(z) 5 n - 1, degAk(z)

3)

ordz=ooHn,q(z) > nm + q.

traversed in the upward direction.

+

5

A2(z)L2(l/z)

n’ n-l,

+.

.. +

An(z)Ln(l/z)

;

for 1 5 k 5 q, forq+l
The linear independence of the numbers in (95) is proved using the ideas in 51.4. A complete set of small linear forms in these numbers is constructed from the values Hn,q(a/b) for q = 0, 1, . . . , m. A result similar to Theorem 2.22, but without the assumption 5 < 0, was obtained in [1982] by Gutnik, using functional approximations of the second kind. In [1986] Vasilenko considered the function

=

(97)

/A.--,I41 +l,...,p+l;z

) ’

and proved a bound depending on the coefficients for a linear form in the numbers 1, Lj(pk, E), where ~1,. . . ,p,. are distinct rational numbers in the interval 0 5 p < 1, for each k the index j varies from 1 to a certain Tk E N, and c is an element of a quadratic imaginary field that satisfies certain conditions. Vasilenko’s proof is based on Hermite-Pade approximations for the set of functions 1, Lj(pk, l/z), 1 5 j 5 rk, 1 5 k 5 r, in a neighborhood of 00. The approximations are constructed in the form of an integral (96), where R(z) is a rational function that depends in some special way on the numbers pk, Tk. The estimates also made use of a representation of this integral in the form of a multiple integral of the type (94). A little earlier Vasilenko [1985d] had obtained a lower bound for linear forms in 1, Lj (j&, [ii) under the assumption that [I,...,& EQ. The paper [Vasilenko 1985d] also contains a bound for linear forms in the values of the hypergeometric function F(a + 1, 1, y + 1; z), a, y E Q, at distinct rational points close to zero. In that case the functional approximations are constructed in the form of an integral

57. Hypergeometric

Series with Finite Radius of Convergence

zn F(cu+l,l,y+n+l;zt)& 27G c ((t - &) . .. (t - &))n+l -1

139

’

where C is a circle of small radius centered at t = 0. A similar construction, leading to functional approximations for the set of functions F( 1, bi, c; z), was announced in [Chudnovsky 198Of]. In [1980b, 198Oc, 198Of] Chudnovsky proposed a general construction of Hermite-Pade approximations of the first kind for generalized hypergeometric functions. Let (~1,. . . , op and pi,. . . , /3, be complex numbers that are not equal to zero or negative integers, and further suppose that none of the differences /3i - oj are equal to zero or negative integers. Then for any n E N there exist polynomials Pi(z) E C[z], i = 0, 1, . . . , q, of degree at most n such that ZPn--l

F

p q

al + qn, . . . , ap + qn pl+pn,...,Pq+pn;z

>

= 2 WZ)fi(Z) i=o

,

(98)

where m,...,ap fob)

= pFq

Pl,...,Pq

;z

> >

andfori = l,... , q the function fi(z) is a generalized hypergeometric function whose parameters are obtained by certain integer shifts in the parameters of fo(z). The A( z > can be expressed linearly in terms of fe(z) and its derivatives through the q-th order. Somewhat different versions of these relations were proposed in [Huttner 19921 and [Nesterenko 19931. In [1986] (see also [1991b]) I vankov proved bounds for linear forms in the values at distinct points of generalized hypergeometric functions and their derivatives in the case p = q + 1. His result is similar to Theorem 2.18, and it uses the same construction that was described in $6.3 for HermitePade approximations of the functions Fj(wkz), 1 5 j 5 m, 1 5 lc 5 t, where FI(z) is the generalized hypergeometric function with parameters and Fj(z) = (pj-1 + z$)Fj-I(Z) for j 2 2. ~l,...,Qp,Pl, .‘., Pq, 7.2. Functional Approximations of the Second Kind. In 1979, generahzing ideas of Maier (see §5.3), Chudnovsky proposed the following method of explicitly constructing approximations for the generalized hypergeometric functions (84) with p = q + 1 that are close to Hermite-Pade approximations of the second kind. For v E Z define

If we denote

P(5z) = (x + al) * * * (z + up), then the above functions

Q(x)

can be written

= z(z + bl - 1). . . (z + b, - 1) , in the form

140

Chapter 2. Effective Constructions

in Transcendental

Number Theory

fv(z)= 2 p(v)Q;;)~(y;(;)-l)Zk. k=O

The proposition that follows [Chudnovsky 1979e].

was proved in a somewhat

more general form in

2.2. Let n E N, V,T E No, and

Proposition

r
WO)

P

Let f”(z) be the function defined by (991, and let P(Z) and Q(Z) I be as above. If A(z) is the degree n polynomial -n,r-n+l,r-n+bI A(z)

=

,...,

r-n+b,

; z-l

znq+2Fp al,a2,...,ap

n

Q(l +;(;;“!

p (ml)j

=

: ;;”

>

:,r - n) zn-j ,

co j=o then the following inequality holds for some polynomial B,(z) at most n - 1: ord,,o(A(z)f,(z) - B,(z)) 2 T + 1 .

E C[z] of degree (101)

Note that the polynomial A(z) does not depend on V. For example, if we choose T = n + [(n - 1)/p] - q, then (100) will hold for all v = 0, 1, . . . , q. Thus, the inequalities (101) show that the set of polynomials A(z), Bo(z), . . . , Bp(z) gives rather good simultaneous functional approximations to fo (z), . . . , f,(z). We shall give a proof of Proposition 2.2 that, while based on the ideas of Maier and Chudnovsky, is somewhat different from the one in [Chudnovsky 1979e]. We define the differential operators D, = yP(y&) and D, = ~Q(z&) acting on the ring C[z, y, z]. For any m 2 0 we have D,(ym) = P(m)y”+l and D,(zm) = Q(m)z m+l. It is easy to verify that the coefficients Ek of the series co f,,(z)A(z)

= 2

Ekzk

k=O

satisfy the following relation for n 5 Ic 5 T:

&=ck0;

(-l)iQ(lc

- n + i + 1) . . . Q(r - n + i).

i=o

* P(i) * 1. P(k + v - n - 1+ i)

= CPY) k-n+yDz)+k

((z

_

(102)

Zy)nZk-n+l

where c = (Q(l)...Q(r - n)P(O)...P(v - l))-‘. Note that there are no Q factors in the sum in (102) when k = T, and there are no P factors when

$7. Hypergeometric

Series with

Finite

Radius

of Convergence

141

k = n - V. It follows from (100) that the differential operator on the right in (102) has order less than n. This proves the proposition. In [1979e] Chudnovsky announced a seriesof number-theoretic applications of his construction of simultaneous functional approximations. However, he did not publish detailed proofs. In [1979b] Nikishin proved bounds for linear forms with integer coefficients in the logarithms of successivenatural numbers. He used the scheme that was described in $1.3, and he constructed simultaneous rational approximations to the values of the logarithm by first constructing Hermite-Pade approximations of the second kind in a neighborhood of 00 for the set of functions fj(z)

= In (1 - q)

- In (I - i)

= l’+’

&

,

j = 0, 1, . . . ) s - 1. His construction generalized the Jacobi identity (see $3.2) in the special case under consideration. If

Q(x) =$(2)n(x"(x -1)" +..(x-sy), .then degQ(z)

= ns, and the functions Rj(z) =

have the representation

where Bj(z) =

i+l &(2)dx x-z Jj

’

j=0,1,...,

Rj(4 =QWj (z) -4(z) ,

7 J‘+lQ(x) -Qua EcIzl j

s-l,

x-z

degBjsns-1.

In addition, ord,=,Rj(z)

> n + 1.

The construction of a complete system of simultaneous rational approximations now proceeds in the same way as in version 4) of the proof of the transcendence of e in $2.2. In [Rhin, Toffin 19861 similar considerations were used to construct Hermite-Pad6 approximations of the second kind for the set of functions ln(1 + akz), k = 1,. . . ,m, at the point 0 and to obtain number-theoretic results from them. See also [Rhin 19871.The same approach was also used by Sorokin [1985], who proved the following theorem. Theorem 2.23. Suppose that (~0,al,. . . , a, are positive rational numbers such that cyk $ q (mod 1) for k > j 2 1, and the finctions @j(z), j = 17”‘) S, are defined by setting #j(z) = F(aj, l,oj+oc; z), vrhere F(o,fi, 7; Z)

142

Chapter

2. Effective

Constructions

in Transcendental

Number

Theory

is the Gauss hypergeometric function. Let < = a/b, where a and b are nonzero integers of a quadratic imaginary field I[. For any E > 0 there exist positive constants 70 and 80 depending only on c, a, b, aj such that if cyola12 < lbl”/(8+1+2”)

thenforanyxs,xi,...,xgEZr

7

with[x1...x,lL&

Ixo+xl~l(~>+~~~+~,~,(~)l~

onehas -l--E

~Xl."%~

s

In the proof, given any vector F = (vi,. . . , ys) E Z”, vj 2 0, one constructs apolynomialQ,E~=[z],degQ~~<_l+...+v,,thatforeachj=l,...,sis orthogonal to all polynomials of degree < Uj on the interval [0, l] with weight D-l (1 - .z)~O-~. Then

d,,,(Q&)@j(l/Z)

- Pjc(z))

> vj + 1,

j = l,...,S)

for certain polynomials PjF(z). If we now choose z = b/a, we obtain the necessary simultaneous approximations to the @j(<). In [1990b] Hata proved a lower bound for linear forms in the values of the functions (97) that was similar to the result in [Vasilenko 1985d] that was mentioned in s7.1. But unlike Vasilenko, Hata used HermitePade approximations of the second kind for the functions (97) with different values of r and p. Let cq,...,(~~ be arbitrary numbers in the interval [0, l), let r1 2 r2 2 .ff 2 r,,, 2 0 be integers, and for each i = 1,. . . , m let ki be the number of indices v, 1 5 v 5 i, for which ov = oi. In [Hata 1990b] it is proved that the polynomial

A(~) =k(-l)j (;) c1+flmn,>...(‘- +jmm “-)xj, j=O

where n = r1 + ... + r,, interval [0, 11:

is orthogonal to the following n functions on the

xj-L+; (in x)k--l,

l
O<j
Since the functions (97) have the integral representation w4

l/z)

= (r -f I)!

’ xp(ln(l/x))‘-l o z-x I

dx

,

it follows that for any j = 1,. . . , m we have the following identity, which is similar to the Jacobi identity (see $3.2): , 1

(r -t l)! I o A(x) where

xrrc1n;,/2)r-1

dx = A(z)L,(p,

l/z)

- B(z) ,

(103)

$8. Remarks

143

B(z) = (r : l)! I ol A’zj I ~(x)xp(ln(l/x))‘-ldx Eaqz]) deg B(z) 2 n - 1. If we let Bj (z) denote the polynomial B(z) that corresponds to p = -oj and T = Icj, expand the left side of (103) in powers of z-l, and use the above orthogonality property, we find that ord,,,(A(z)&

(-cyj,

l/z)

- Bj(z))

>_ rj .

After substituting z = q E % in our identities with parameters aj E Q and IqI sufficiently large, we obtain simultaneous rational approximations to the numbers Lkj (-oj, l/q), and, using the ideas in $1.3, a lower bound for linear forms in these numbers. The paper [Hata 1990b] also gives a number of corollaries of the general theorem, for example, concerning linear independence of the polylogarithms of numbers of the form &l/q. Finally, we note that the polynomial A(z) in [Hata 1990b] can also be written in the form

$8. Remarks In this chapter we have described far from all of the results in transcendental number theory that can be proved using explicit constructions of functional approximations. In this section we will briefly mention some of the results that have been omitted. 1) In [1932] Siegel used an identity similar to (41), but with et replaced by Q(Z), to prove the following: Let p(z) be the Weierstrass elliptic function with periods wi and ws and invariants gs and gs. If g2 and gs are algebraic, then at least one of wi, w2 is transcendental. Siegel’s technique of proof is close to the method of Gel’fond that we described in 54.2. Somewhat later, Schneider proved that both periods wi and wp are transcendental, but he used a different method. His work will be described in Chapter 3. 2) In [1921] Tschakaloff M c

(-pa--n(n+l)/2

used Pad6 approximations >

a

E I[,

o! #

0,

to prove irrationality

a E ZI,

Ial

>

1 ,

of

(104)

n=O where ll is a quadratic imaginary about values of the functions

field. He also proved a more general fact

144

Chapter 2. Effective Constructions

W(2)= 2

in Transcendental

pa--n(n+l)/2

Number Theory

a E 251.

7

n=O

Namely, he showed that if (~1, . . . , (Y, E 1, ak # 0, ak/% # at, t E z, k # T, then the numbers l,w(oi), . . . , w(cr,) are linearly independent over I[. A lower bound for the absolute value of a linear form in 1, w(LY~), . . . , ~(a,) was obtained by Stihl [1984]. In [1943] Lototskii studied the values at points of a quadratic imaginary field ll of the function fa(z)

= fi

(1+ zKn)

)

a E Cc, ]a] > 1 ,

n=l

which

satisfies

the functional

equation fa(az)

He proved that if a E Zr, used the method described In the 1970’s and 1980’s relating to special cases of

= (I+

z)fa(z)

.

cx E II, and ofa # 0, then f(o) $? I[. His proof in $4.2. Bundschuh and Walliser proved several theorems the equation

daz)

= P(z>cp(z)

+

Q(z) ,

(105)

where a and the coefficients of the polynomials P and Q are chosen in a quadratic imaginary field Il. They studied the values of p(z) and its derivatives at points of Il. In particular, Bundschuh [1969] estimated the irrationality measure of fa(a) under certain conditions, and Walliser [1973b] generalized Lototskii’s result to arbitrary equations of the form (105), where P(z) = az + b. The irrationality of the numbers in (104) is a consequence of Walliser’s theorem. Between 1986 and 1991 Popov obtained several results in this area (see [Popov 1986, 19901). A mong those working on these questions he was the first to pass from quadratic imaginary fields to arbitrary algebraic number fields. In particular, he obtained bounds for approximations of fa(a), fa(a)/fa(P), and fA(a)/fa(cr) by algebraic numbers; and he also proved inequalities for the number of linearly independent values over II among the numbers F(o), F(aa), . . . , F(aa’+l), where F(z)

= 1+ 2

n=l P(a)P(a2)

Zn

. . . P(a”) ’

P E Z[z] .

He proved the latter result by a method of elimination of variables that is different from the techniques described in 31. Namely, he used the following fact (see [Nesterenko 1985b]): Suppose that cl, cs, ri,rs are positive numbers, 0 5 71 - 72, c(t) is a monotonically increasing function on N, and

58. Remarks

lil&(T(t)

145

. 4t+ 1) 1 t4Eacr(t) = .

= +cx3,

Further suppose that 3 = (01,. . . ,0,) E IWm, and for all N E N sufficiently large there exist linear forms LN(~) = ai21 + . . . + umz,, aj E Z, such that H(LN)

cle +locN) < lL~(Zj)l 5 c2e--TZU(N)

5 eotN),

Then there are at least (71 + l)/(l + 71 - 7s) linearly independent numbers among0i,...,9m. We note that Popov also proved results about the values of functions that satisfy equations of the form m-1 f(a”z)

=

c

Pk(4f(akz)

+

P-l(Z),

k=O

His proof was based on a method that will be described in Chapter 3. 3) We give one more example, due to Derevyanko [1989], of the use of explicit constructions of Pad6 approximations. Let f(z) = gCg12L ) k=O

where {ck} is a sequence of fl’s. In [Derevyanko 19891it is proved that, if I[ is either Q or else a quadratic imaginary field, and if b,p, q E Zr, lb] > 1, then

In the proof Derevyanko constructs polynomials Pn(z) and Qn(z) of degree 5 n for which Qn(z)f(z)

- p,(z)

= z2n+1 +

5

bkzk .

k=2n+2

It turns out that the coefficients of the Qn(z) and also all of the bk are equal to 0 or fl. If II = Q, then a similar result follows from the continued fraction expansion of f(l/b) in [Langevin 19921and [Shallit 19791.

146

Chapter 3. Hilbert’s

Hilbert

Seventh Problem

Chapter 3 ‘s Seventh Problem

51. The Euler-Hilbert

Problem

1.1. .Remarks by Leibniz and Euler. Leibniz stated that “the number (Y = 2r/“, a = 4, is intercendental,” but he did not explain what the term meant, let alone give a proof. In Euler’s introduction to analysis [Euler 19881, when discussing the properties of the logarithmic function with rational base a, he wrote that “unless the number b is a [rational] power of the base a, the logarithm of b cannot be expressed as a rational number.... It is especially desirable to know the logarithms of rational numbers.... Since the logarithms of [rational] numbers which are not the powers of the base are neither rational nor irrational [algebraic, in modern terminology], it is with justice that they are called transcendental quantities. For this reason, logarithms are said to be transcendental.” Like Leibniz, Euler did not give a proof of his statement; in fact, it seems that nothing was published on this question before the twentieth century. 1.2. Hilbert’s Report. Hilbert gave his famous report on “Mathematical Problems” in August, 1900 at the second International Congress of Mathematicians. Noting the “profound value which definite problems have for the progress of mathematical science,” he gave a list of 23 problems (‘the study of which could greatly stimulate the further development of our science.” Hilbert’s seventh problem was titled “irrationality and transcendence of certain numbers.” We again quote from Hilbert: When we know that certain special transcendental functions that play an important role in analysis take algebraic values for some given algebraic values of the argument, this circumstance strikes us as especially surprising and worthy of further study. We always expect that transcendental functions generally take transcendental values at algebraic values of the argument. And although we know that there even exist entire transcendental functions that take rational values at all algebraic numbers, we still consider it highly likely that a function such as eKit, which obviously takes algebraic values for all rational z, will nevertheless take transcendental values for all irrational algebraic z. This statement can be given a geometrical character as follows. If the ratio of vertex angle to base angle in an isosceles triangle is an algebraic but irrational number, then the ratio of the base to one of the lateral sides is a transcendental number.* Despite the simplicity of this statement, and its similarity * To see the equivalence of the two forms of Hilbert’s conjecture, let a denote the base angle of the isosceles triangle, so that the vertex angle is X -2a. Then the ratio of angles is (r/o) - 2, which is rational or algebraic if and only if

§2. Solution of Hilbert’s

Seventh Problem

147

with problems solved by Hermite and Lindemann, it seems to me to be extraordinarily difficult to prove it, just as it seems to be to prove that c@ is always transcendental (or at least irrational) if (Y is algebraic and /I is irrational and algebraic, for example: 2& or ek = im2j. It might be true that the solutions of this and similar problems require us to develop new methods and new points of view on the essential nature of special irrational and transcendental numbers. It is interesting to note a further statement by Hilbert on the difficulty of his seventh problem. Siegel recalls (see [Siegel 19491, p. 84) that Hilbert would often say that the proof of the irrationality of 2& belongs to the more distant future than the proof of the Biemann Hypothesis or Fermat’s Last Theorem. Although here Hilbert was mistaken, he was certainly correct when he predicted that his seventh problem would be a great stimulus to research. Its solution led to the creation of a powerful new analytic method, which is being applied and further developed even in our day. This method has made it possible to solve many problems that could not be attacked by earlier methods.

$2. Solution

of Hilbert’s

Seventh Problem

2.1. Statement of the Theorems. We give several equivalent formulations of the theorems that resolve the question in Hilbert’s seventh problem. Theorem

3.1. Let a, b E A, a lna # 0, b # Q. Then ab is transcendental.

Theorem

3.2. Let (Y,/3, y E A, C@In p # 0, y = In (Y/ In /?. Then y E Q.

In fact, if y were an algebraic irrational, then by Theorem 3.1 the number cy = @7would have to be transcendental. Theorem 3.2 can also be stated as follows. Theorem 3.3. Let (Y,p E A, a/l # 0. If the numbers lncu and lnfi are linearly independent over Q, then they are linearly independent over A.

Clearly, Theorem 3.2 contains Euler’s statement as a special case, and Theorem 3.1 contains Leibniz’s assertion. Theorem 3.2 also implies that if the logarithm of an algebraic number to an algebraic base is irrational, then it is transcendental. 2.2. Gel’fond’s

Solution.

Theorem 3.2. Let cr,p,y

(See [Gel’fond 1934a, 1934b].) We want to prove E A. Suppose that q E N and

f(z) = f:

ck,lew”a+‘w~

)

k,l=O

z = o/r is. But the ratio of sides of the triangle is 2 coscr = eiQ - emia = erri - (eriZ) -‘. This number is transcendental if and only if exiz is.

148

Chapter 3. Hilbert’s

where the Ck,l are constants f(“)(z)

Seventh Problem

in Z. Obviously

= 2

Ck,l(Llna

+ Iln/3)Se(k’““+““P)Z

.

k,l=O

Further

set

fs,, = f’“‘(x) h-“,6 = 2 ck,l(ky+ l)s(Ykzp’z

(1)

k&O

for x E z; x0 = [ln2 In 471,

f2(A,B) One can use Theorem

SO

= [lnq~~,nq] 7

= {(s,x)

: s, x E No,

s 5 A, x < B} .

1.9 to show the existence Ifs,,1 I e-

C = kw(q2)1 , of ck,l E Z such that

xoq2 In q/ In In q

7

(2)

where o<maXICk,ll

(5 x> E mso, x0),

SC.

Here and later Xc, X1, X2, . . . denote positive constants that do not depend on 4. The number fS,z is the value of a polynomial with rational integer coeffi-’ cients at the algebraic point (cr,p, y). If fS,s # 0, then, using (l), we see that the corresponding inequality in (10) of Chapter 1 takes the form Ifs,zl 2 e--xlqz, For q sufficiently f+

(%X) E fqso,zo>

large the inequalities = f’“‘(x)

= 0,

*

(3)

(2) and (3) are incompatible;

(%X1 E ~(so,xo),

hence,

4 2 cl0 .

This implies that p(z) = f(z)

fi(z x=0

- x)-1-so

is an entire function. Using the maximum principle, we find an upper bound for Iv(z)] and then If(z)] on the disc ]z] 5 fi. If we now use Cauchy’s formula for the derivatives, we find that y5aa

If(“)(z)1

< e-“.3q2’n1nq,

From this and from (11) of Chapter fS,% = f’“‘(x)

= 0,

(s,x)

s<so,

Q~Q12Qo.

1 we obtain: E fin(so, da,

q 2

42

L

Ql

.

$2. Solution

of Hilbert’s

Seventh Problem

149

We now have so many zeros of f(z) that, using the maximum principle, Cauchy’s theorem, and (l), we obtain the inequality Ifs,01 = If(“)(O)/

In’ /3] 5 e-x2qQ’4

for s = O,l,. . . , (q + 1)2 - 1, q > q3 2 42. If we compare this with the lower bound for the numbers fs,s obtained from (11) of Chapter 1, we see that all of these numbers must be zero. We thereby obtain a system of homogeneous linear equations in the ck,l: fs,O

=

f:

ck,l(h+l)’

=

0,

s =

0,.

. . , (4+l)2-l,

Q2

44

L

43

* (4)

k,l=O

By (2), not all of the ck,[ are zero; hence, the system (4) has determinant zero: s = (q + 1)s - 1 . [(k-f + z)“l k;~~“,.,.,.;~~ = 0, Since this is a Vandermonde determinant, it vanishes if and only if two of the ky + 1 are equal. Thus, hy+h

= ksy+

l2,

{h,h)

#

{~2,Z2},

h,k2,h,Z2

E N,

and this implies that y E Q. This success resulted from a skillful combination of methods of number theory and analysis. First, number-theoretic techniques were used to construct, an entire function f(z) $0 that does not increase very rapidly and has a large number of zeros in a certain disc. Next, analytic considerations enable one to show that f( z ) is small in a certain disc of large radius. Then this allows one to use a number-theoretic argument to obtain new zeros of f(z), and so on. As a result one ends up with too many zeros. It would be possible to finish the proof in another way. We could continue indefinitely accumulating zeros (this could be formalized using induction), and prove that f(“)(O) = 0 for all s E No, so that f(z) E 0. Since the ck,l are not all zero, it follows that at least two different k In cx+ 1In ,0 are equal, and hence ~=~tQ. The above method - and also an improved version that enables one to estimate the transcendence measure of certain numbers - are usually called “Gel’fond’s second method.” 2.3. Schneider’s Solution. In [1934a] Schneider proved Theorem 3.2 independently, but somewhat later than Gel’fond. Again suppose that (Y,/I, y E A, y = In a/ In ,6, and let wr , . . . , w, be a basis of the field K = Q(Q, p, y). Schneider considers a different auxiliary function:

Chapter 3. Hilbert’s Seventh Problem

150

&)

= c

$7 2n-1 c

k=l

Ck,l@k=Z1

= f:

l=O

pkzdz>

7

k=l

where n, q E N, q = 10 + 6~. Let a, b E Z. Then for ck,l

= ck,l,lwl

+

with

* '* + ck,l,&

the numbers

ck,l,j

E z

Q 2n-1 cp(a

+

b-f)

=

c

c

k=l

1=0

Ck,labkpak(,

+

b)’

are the values of polynomials over Z at the algebraic point (a, p, y, wi , . . . , w,). Using the Dirichlet pigeonhole principle (Theorem 1.9), one can find a nontrivial set of numbers ck,l such that da + b) = 0,

a,b=

l,...,

&ii,

ck,l

E &Z

.

If we use the maximum principle and Theorem 1.11, we can enlarge the set of zeros of p(z). As a result we obtain (P(a + b) = 0,

a, b = 1,. . . ,3&i

.

By examining the corresponding determinant, one can show that the numbers ck,l must be zero. This contradiction completes the proof of the theorem. The above methods of Gel’fond and Schneider have much in common. They are essentially variants of the same method, the main point of which is to construct auxiliary functions with a large number of zeros in a certain disc, and then enlarge the set of zeros using a combination of number-theoretic and analytic considerations. The main difference between the two methods is in the form of the auxiliary functions: in Gel’fond’s scheme one is using only a linear system of zeros (z E NO), whereas in Schneider’s scheme the zeros are two-dimensional, i.e., they depend on two integer parameters a and b, so that Schneider’s function has many more zeros in a given region. On the other hand, Gel’fond’s zeros have high multiplicity, while Schneider has only simple zeros. The point is that for s > 0 the number PC”)(a+ ky) is a polynomial with algebraic coefficients in the transcendental number In p, whereas the numbers f(“)(x)/ In” p are algebraic. 2.4. The Real Case. If cr, p, and 7 are real numbers, then Theorem 3.2 can be proved without using the theory of analytic functions. Such a proof was given in [Gel’fond 19621. Instead of the maximum principle one uses Lemma 3.1. Suppose that the real-valued function f(x) is continuous on the interval [a, b] along with its derivatives through the n-th order. If f (x) has n or more zeros on [a,b] ( counting multiplicities), then u I

n,

a<x
$2. Solution

of Hilbert’s

Seventh Problem

151

The proof of Theorem 3.2 proceeds along the samelines as Gel’fond’s earlier proof, except that at the last stage one applies Lemma 3.2. Suppose that (~1,. . . , a, E IR, PI (x), . . . , P,(z) f(z)

= g

e-Pk(Z)

E IR[z], and

q-i0 .

k=l

Then the number of real zeros of f(z)

( counting multiplicities) is at most

degPi +..a +degP,+m-1. These lemmas are also used in [Gel’fond 19621to prove Lindemann’s theorem on transcendence of ea for a E An R, (Y # 0. 2.5. Laurent’s Method. In [1991] Laurent proposed a new technique for proving transcendence of certain numbers (see also [Laurent 1992b] and [Laurent, Mignotte, Nesterenko 19951). This method has also been used for other purposes, such as finding bounds for linear forms in the logarithms of algebraic numbers and giving a new proof of Roth’s theorem (we already mentioned this in 57.3 of Chapter 1). We shall give the basic ideas of the method and its applications using the example provided by the Gel’fond-Schneider theorem. But we shall not concern ourselves with giving the exact values for the parameters. Let (Y,,8, y E A, y = In p/ In cr. Consider the matrix

(b + YY)kcw”)OQ,y<X,

O
Oll
’

where the pair (k, Z) determines the column and the pair (z, y) determines the row. If the number X2 of rows is much more than the number KL of columns, then we can use theorems bounding the number of zeros of the function

to show that there exists a set of KL rows of the above matrix which form a KL x KL matrix with nonzero determinant

A = 1(x +

(..~)Ew O
yy)“&+“I

where

w c {bc,Y>: o
card(W) = KL = it4 .

O
Let h,l(z)=z

We introduce a new variable C, and set

k

f2’

1.z

.

152

Chapter 3. Hilbert’s

Seventh Problem

D(C) = If((x + -d

1o<~;&)yv
Then D( 1) = A. The function D(C) has a zero at C = 0 of order at least KL(KL - 1)/2, as follows from the fact that the derivatives D(“)(C) ICC0 are sums of numbers that are proportional to determinants of the form

and these determinants are nonzero only when all of the aj are distinct. Thus, they are nonzero only if s 2 0 + 1 + . . . + (M - 1). The maximum principle now enables one to obtain an upper bound for ID(l)1 = [Al. Meanwhile, A is the value of a polynomial over Z at the algebraic point (a, /3,r); by Theorem 1.5, this gives a lower bound for [Al. With an appropriate choice of the parameters K, L, and X the upper and lower bounds are incompatible. This contradiction completes the proof.

$3. Transcendence

of Numbers Connected Functions

with Weierstrass

3.1. Preliminary Remarks. We recall some of the properties strass functions Q(Z) and C(z). Let Wl,W2

E @,

The Weierstrass

7 =w1/w2

@function

n={2zwl+2yw2

4 R

of the Weier-

: z,yEz}.

is defined as

.dz)= f + g ((z Yw)’ - f) ’ where C’ here and in the sequel means that the sum is taken over nonzero w. The function C(z)=f+C’(-&+$+-g WER

(6)

is called the Weierstrass C-function corresponding to p(z). We clearly have p(z) = -C’(z), and g(z + w) = p(z) for w E 0. The set 0 is called the period lattice of g(z). The series (5) and (6) converge for all z $! R. We have G42

= 4dz13

-

g2dz)

-

Q3

7

(7)

where g2 = 60 c’ w-~, WEf-2

g3 = 140 c ’ w+ WEQ

;

and C(Z + 2Wj) = C(Z) + 77j,

Vj = 2C(wj),

j=1,2.

$3. Transcendence

of Numbers

Connected

with Weierstrass

Functions

153

The numbers ~1 and ~2 are called the quasiperiods of C(z) corresponding to the fundamental periods 2wi and 2~2 of p(z). It is clear that the fundamental periods of p(z) (i.e., basesfor the lattice 0) can be chosen in infinitely many ways. The numbers gs and gs are called the invariants of p(z). In Gel’fond’s and Schneider’s proofs that we discussedin $2, an important role was played by the addition property of the function ez (i.e., euf” = e”.e”), which meant that algebraicity of e” implies algebraicity of eaz for z E Z, and by the fact that et is an entire function that does not grow very rapidly. In Gel’fond’s proof an important role was also played by the differential equation for et (i.e., y’ = y), which enabled us to use the derivatives of eaz. In an analogous fashion, the function p(z) satisfies the differential equation (7) and the following addition laws (where U,O,U + w $ R, @J(U)# g(u) in the first formula, and 221$! L? in the second formula):

From (7) and (8) it follows that if gs and gs are algebraic numbers, then ‘algebraicity of p(a) implies algebraicity of p(‘) (ma) for s E No, m E Z, mo $ R. Thus, the only thing standing in the way of using the methods of $2 is that p(z) is not an entire function (it has double poles at the lattice points). But it turns out that it is possible to get around this obstacle. A technique to do this was already being used by Siegel in his work on the periods of an elliptic function that we mentioned in Chapter 2. 3.2. Schneider’s Theorems. In [Schneider 193413,19571 several important theorems are proved about values of the functions g(z), C(z), and e’. Theorem 3.4. Let g2 and gs be the invariants of p(z), function corresponding to p(z), and let P(Z) = az + K(z),

I4 + PI > 0, a,b,B E C

let (‘(z) be the C-

P 4 fl .

Then at least one of the numbers 92,

93,

a, 6

!ia%

w

(9)

is transcendental. Theorem 3.5. Suppose that fi,q E C:, the Weierstrass functions p(z) and ~1 (qz) are algebraically independent, and 92, g3 and h2, h3 are their invariants, respectively. If j3 is not a pole of these functions, then at least one of the numbers 92, 93, ha, h3, dP>7 m(d% Q is transcendental.

154

Chapter

Theorem

3. Hilbert’s

Seventh Problem

3.6. Suppose that p(z) is the Weierstrass function with invariants

g2 and g3, ,B $ 0, q # 0. Then at least one of the numbers g2,

g3,

dP),

9, eqp

is transcendental. We now give some consequencesof these theorems. Theorem

3.7. If g2, gs E A, then ~1, ~2, q1 and r/2 are transcendental

num-

bers. Theorem 3.8. If g2, g3 E A and j3 # 0, then at least one of the numbers p and p(p) is transcendental.

Theorem 3.8 is an analogue of Lindemann’s theorem. It can be reformulated in the following ways. Theorem

3.8’.

If g2, g3, /J E A, /3 $ 0, then p(p) is transcendental.

This is analogous to saying the e@is transcendental. Theorem

3.8”.

If g2,g3, p(p) E A, then p is transcendental.

This is analogous to saying that if (Y E A, CE# 0, then In (Yis transcendental. Corollary 3.1. If the semiaxes of an ellipse are algebraic numbers, then the circumference of the ellipse is transcendental. Theorem 3.9. Let g2,gs E A and /3 4 R. Then p(p) and C(p) cannot both be algebraic numbers. The same is true of g(p) and <(/3)//?. Theorem 3.10. Let g2, g3 E A. Then each of the triples ~1, ~1,l and w2,772,1 is linearly independent over A. 3.11. Let g2,g3 E A. Then the numbers wl/ql, + 772)are transcendental.

Theorem

wz)/(vl

w~/Q, and (~1 +

Theorem 3.12. Suppose that the functions p(z) and ~1 (qz) are algebraically independent, and their invariants are algebraic numbers. If p(p) E A, then q and pl(q/3) cannot both be algebraic. Theorem 3.13. Let g2, gs E A. If (Y,p(p) E A, (Ylntv # 0, then p/ lncu is transcendental. In particular, W/K is transcendental for any period w of p(z). Theorem 3.14. Let g2,g3 E A. If (Y,p(p) E A, then ep and g(lna) are transcendental numbers. In particular, p(ni) and ew are transcendental, where w is a period of p(z). Definition 3.1. A number X is called a multiplier of the lattice 6? (or of the function p(z)) if XR c 0. Clearly all rational integers are multipliers. It is known that g(z) has irrational multipliers if and only if T = wl/wz is a quadratic imaginary irrational number. In that case one says that p(z) has complex multiplication. All of the multipliers then lie in the field Q(T).

$3. Transcendence

of Numbers

Connected

with Weierstrass

Functions

155

Theorem 3.15. Let g2,g3, p(u), p(w) E A. If p(z) does not have complex multiplication, then u/v is either rational or transcendental. If p(z) has complex multiplication, then either u/v E Q(r) or else u/v is transcendental. This theorem is an analogue of Theorem 3.2. If r E Cc, Imr > 0, and R is the lattice spanned by 1 and 7, then the gs = 92(r) and gs = gs(r) become functions of 7. The modular invariant j(r) is defined by setting invariants

92(T)3

j(~) = 1728

-

92(T13

L73(TJ2

.

As a function on the complex upper half-plane, j(r)

satisfies the identity

for all integers a, b, c, d with ad - bc = 1, i.e., it is a modular function. It is well known that j(r) is algebraic if 7 is any quadratic imaginary number. It follows from Theorem 3.15 that j(r) is transcendental if r is an algebraic number that is not quadratic imaginary. Since the modular invariant function has period 1, it has a Fourier expansion j(T) = J(q) = q-l + 744 + 2

c,qn,

GXEZ,

n=l

where q = e2xiT, that converges in the complex upper half-plane. In [1969] Mahler conjectured that J(q) is transcendental for any algebraic value of ‘q, 0 < IqI < 1. This conjecture was proved in [Barre-Sirieix, Diaz, Gramain, and Philibert 19961. 3.3. Outline of Proof of Schneider’s Theorems. Let us consider Theorem 3.4. Suppose that all of the numbers in (9) are algebraic. Let

K = Q(a, byaa, dP>, d(P), VW) , Y = deg K,

g = 12V + 1,

n E N, f(z)

=

2

Ck,Wk(4&)

r = [n2/2gu] *

) (10)

k,l=O

We remove all the points in the set {m/I : m E Z} that belong to 0, and renumber the remaining points in the order of increasing absolute value: The function C(z) has addition law Pl,PZ,....

156

Chapter

3. Hilbert’s

Seventh Problem

C(2z) = 2<(z) + ;$# From

(12)

(12) it follows that

(7), (8), (11) and

(PWn),

.

63Wn)

for

EK

KEEN, 5ENf-J.

Using the Dirichlet pigeonhole principle and the inequality (11) of Chapter 1, one can prove that there exist ck,l and y such that ck,l

E z,

0 <

P(Pn)

=

o
ick,ll

K.= l,...

0,

5

,g;

+f

,

s=0,1,...,

(13)

r-l.

Suppose that there exists t > T such that f’“‘(PK)

/c= l,...

= 0,

,g;

s=0,1,...,

t-1;

f’“‘(Pj)

# 07

say, for j = 1. Since c(z) has simple poles at the points of R and p(z) has double poles there, it follows that T f(Z)

n

(z

-

233.4

-

2y43n

=

TEN,

f(Z)PT(Z),

x,y=-T

is a regular function on the closed parallelogram LT having corners at (2T + l)(fwr f ws). The following function is also regular on LT:

f(Z)PdZ) Q(z) ’ Let T = [t’l”],

Q(z) = fi(z

where

- P$

.

n=l

and let 1~ be the boundary of LT. We have t! 2?ri

f(z)PT(Z)

f lT Q(z)(z

& -

/31)

f(Z)PdZ) =

Q(z)

‘! ?%

= PT(~l)f(t)(~~) Ql(Pd

’

where

Ql(z) = Q(z)(z -PI>-”

.

From this we derive the inequality If’4(pl)

15 &2-d’-Yte-d

,

where 71 is a constant. Since, by assumption, f(“)(Pr) Theorem 1.5 that

(14 # 0, it follows by (15)

$4. General Theorems

157

where 7s is a constant. Since g > 12v, it follows that for t sufficiently large i.e., for a sufficiently large parameter n - the inequalities (14) and (15) are incompatible. This means that for n sufficiently large we have f(“)(Pr) = 0 for all s E No. Thus, f(z) E 0. But it is well known that the functions p(z) and Q(Z) are algebraically independent. Hence, all of the ck,l vanish, and this contradicts (13). The proofs of Theorems 3.5 and 3.6 are similar. One need only replace v(z) by PI(V) or eqz in (10); and at the last stage of the proof one needs algebraic independence of p(z) and @l(qz) or of p(z) and eqt.

94. General Theorems 4.1. Schneider’s General Theorems. Definition 3.2. Let jr(z) and fs(z) b e entire functions of order hl and h2, respectively, and let H = max(hr, h2) and F(z) = fr(z)/fs(z). By the order of the meromorphic fin&ion F(z) we mean inf (II) taken over all representations of F(z) as a ratio of two entire functions. We also recall that for a E A we denote m = max loti) 1, where the maximum is taken over all conjugates of cr. All of the earlier theorems in this chapter are special casesof the following general fact, which Schneider proved in 1949. Theorem 3.16. Let f(z) and g(z) be either entire or meromorphic functions of order no greater than p, and let ~1,. . . , z, E Ccbe distinct from one another and from the poles of the two functions. Further suppose that the numbers f’“‘(zj),gyzj), j = 1,. . . ,m, s = O,l,. . ., lie in an algebraic number field K of degree n, and there exist constants a > 0 and b E N such that bs+‘f@)(zj),

If@)(zj)l,

b’+‘g@)(zJ

E ZK ,

)g(s)(zj)) 5 b’+‘(s + l),’

.

If m > (2~ + 1)(2na + n - 1 + 0.5) , then f(z)

and g( z ) are algebraically dependent.

The proof of this theorem uses the same ideas as the proofs of Theorems 3.2 and 3.4. We first construct an auxiliary function @5(Z)= 5 k=O

9, Ck,lf’(z)g’(z) I=0

.

Chapter 3. Hilbert’s

158 Using the Dirichlet

pigeonhole

principle

Seventh Problem and the inequality

(11) of Chapter

1,

we prove that there exists a nontrivial set of not very large integers ck,[ for which &‘(Zj)

= 0,

j=l,...,

m,

s=0,1,...,

SI-J.

We then use the maximum principle and (11) of Chapter 1 to successively enlarge the set of values of s. If necessary, at this point we can compensate for the poles of f( z ) and g(z) by a factor similar to the one in the proof of Theorem 3.4. In the end we are able to prove that G(z) E 0. Since the coefficients Ck,l are not all zero, this means that the functions f(z) and g(z) are algebraically dependent. Theorem 3.17. Let f(z) and g(z) be either entire or meromorphic functions of order no greater than p that satisfy differential equations of the form y(“) = P(y, y’, . . . ) y(“-l)),

P(u,,

. . . ,w-1)

E ~[~O,.~.,wA]

.

Suppose that all of the numbers f’“‘(z&

g(s)(z.) 3 7

j=l

>***, m,

s
where the order of the derivative of each function is less than the order of the corresponding diflerential equation, and all of the coeficients of the polynomials P lie an an algebraic number field of degree n. If m > (2~ + 1)(3n - 0.5) , then f(z)

and g(z)

are algebraically

4.2. Consequences of Theorem the theorems in 53 from Theorem

@3),

dependent. 3.17. We now show how to derive some of 3.17.

1) We prove the simplest version of Lindemann’s theorem. Let f(r) = z and g(.r) = et. These entire functions obviously satisfy the differential equations y’ = 1 and y’ = y, respectively. Clearly p = 1. If cr and ea are algebraic, then so are tcu and eta for any t E Z; in fact, all of these numbers lie in the field Q(o, ea). Let n be the degree of this field. If we take m > 9n, t = 1,. . . ,m, then all of the conditions of Theorem 3.16 are fulfilled, and so the functions z and eZ are algebraically dependent. Hence for nonzero (Y E A the number e” is transcendental. In the case (Y = 1 this is Hermite’s theorem. 2) To prove Theorem Zj=jlna,aEA,alna#O.

3.1 we take f(z)

= e+, g(z) = ebz, b E A, b # 0,

3) To prove Theorem 3.4 we take f (2) = p(z) = az + bC(z), g(z) = p(z). For these meromorphic functions p = 2. These functions satisfy the differential equations Y

11, =

--

6 by

and

y” = 6y2 - ;gz ,

(17)

§4. General Theorems

159

respectively. The relations (7), (8), (11) and (12) allow us to construct an infinite sequence of points zj = /?j so that (16) is fulfilled. But v(z) and Q(Z) are algebraically independent (as we already mentioned in 53.3). 4.3. Lang’s Theorem. Various authors have proved theorems that resemble Theorems 3.16 and 3.17. We give one such theorem, belonging to Lang (see [Lang 19841). Theorem 3.18. Let K be an algebraic number field of degree n, and let , fN(z) be meromorphic functions of order no greater than p. suppose fib),... that the field K( fi , . . . , fN) has transcendence degree greater than 1, and suppose that the ring lK[fi, . . . , fN] is closed under differentiation. If WI,. . . , w, are distinct complex numbers and i = l,...

fi(Wj) E K

,N,

j=l,...,

m,

then m 5 10pn. Theorem 3.18 contains Theorem 3.17, except that (16) is replaced by a somewhat stronger condition. As an example of its applications, we note that Lindemann’s theorem is an easy corollary. Namely, suppose that a and ea are ,algebraic numbers in a field IK of degree n. Let fi(z) = z and fi(z) = et. These functions are algebraically independent over IK, and the ring K[z, ez] is closed under differentiation. Obviously p = 1. For any t E Z the numbers ta and et0 belong to K. But then the numbers CY,2a,. . . , (10n + 1)a cannot all be distinct; hence, Q = 0. 4.4. Schneider’s Work and Later Results on Abelian Functions. DeEnition 3.3. Let y = y(x) be an algebraic function, i.e., a solution of the equation

V%V> E a44

WX,Y> = 0,

P(%V) $0 *

Integrals of the form

are called abelian integrals. Examples of abelian integrals include

JJ

dt

4t3 - gyt - g3 ’

which arises when one inverts the Weierstrass @function, and the Euler betafunction 1

Bbq)

=

J 0

xp-‘(1

- x)q-ldx

7

P,QEQ,

P,q>O*

(18)

160

Chapter

3. Hilbert’s

Seventh Problem

In [1941] Schneider applied the method in this chapter to functions of several complex variables in order to study the values of abelian integrals and abelian functions. As a consequence of his general theorem he proved that B(p, q) is transcendental for p, q E Q, p, q 4 Z. Subsequently other results were proved on the values of abelian functions and integrals. We now mention some of them. In [1977a] Masser proved that the five numbers B(1/2, k/5), k = 1,2,3,4,5, are linearly independent over A. In [1980] Laurent proved that if a, b E Q, a, b $! Z, N is a common denominator of a and b, H > ee, and M = max(l,(p(N)/2), where cp is the Euler p-function, then there exists an effective constant c = c(N) > 0 such that for anvtEAonehas IB(a, b) - (1 2 e-cdMT(lnT)M

,

where d = deg<,

H(C) L H,

T=lnH+dlnd.

Consider the Gauss hypergeometric function ca a(a+l)...(a+n-l)b(b+l)...(b+n-l)z, F(u, b, c; z) = 1+ c n!c(c + 1) . . . (c + n - 1)

>

n=l

c # 0, -1, -2,. . .) with rational parameters a, b, c. In [1988] Wolfart proved that, if one excludes certain exceptional triples a, b, c E Q and certain corresponding algebraic values of z, then F(u, b, c; z) is transcendental for all other a, b, c E Q and z E A. The exceptional 4-tuples (a, b,c,z) are described in [Wolfart 19881. It should be noted that these exceptional cases include not only the obvious algebraic hypergeometric functions, such as (1 - 2)-l/2

= F(-l/2,1,1;

z) ,

but also sometranscendental functions. For instance, F(1/12,5/12,1/2; z) and F(1/12,7/12,2/3; z ) are transcendental functions, but in [Beukers, Wolfart 19881it is shown that 12’ 12’ 2’ 1331

+ii

and

F(&;;~)=+.

In this connection see also [Flach 19891and [Joyce and Zucker 19911. In [1988] Wolfart takes advantage of the fact that the values of the hypergeometric function are related to the periods of abelian integrals, and uses a general theorem of Wiistholz [1984c] on algebraic groups. Wiistholz’s theorem generalizes such classical results of transcendental number theory as Lindemann’s theorem, and implies that certain classesof abelian integrals are transcendental. In this connection we should also cite papers by Bertrand [1983], Beukers [1991], Cohen [1993, to appear], Cohen and Wolfart [1993], Hirata-Kohno

$5. Bounds for Linear Forms with Two Logarithms

161

[1992], Holrapfel [1993], Laurent [1983], Masser [1977a], Masser and Wiistholz [1990-19951, Morita [1972], Shiga and Wolfart [1995], Wolfart [1981, 1983, 1984, 1985/1986, 19871,Wolfart and Wiistholz [1985], and Wiistholz [1986]. In [Masser 19761and [Coates and Lang 19761estimates are proved for linear forms in algebraic points of abelian functions. Masser’s paper also contains applications of his results to certain Diophantine problems. See [Bertrand 19871for a survey of the results in this active area of recent research and their application to Diophantine geometry.

$5. Bounds for Linear Forms with Two Logarithms 5.1. First Estimates for the Transcendence Measure of ab and lncu/ Ino. The method used to solve Hilbert’s seventh problem turned out to be suitable for obtaining quantitative results as well. We begin with the following equivalent problem (see (8) of the Introduction): estimate the difference between these numbers and algebraic numbers. The first two theorems of this type were proved by Gel’fond in [1935]. Theorem 3.19. Let a, b E A, aln a # 0, b 4 Q, n E N, E > 0. There exists an effective constant Ho = Ho(a, b, n, E) such that for any c E A with deg< 5 n and H(C) > Ho one has

lab - H-(‘“‘“H)6+E,

H = H(C) .

(19)

Theorem3.20. Leta,~~A,~~lncrln~#O,~=ln~/ln~,~~~,n~~, E > 0. There exists an effective constant HI = Hl(a, p, In CY,ln/3, n, E) such that for any C E A with degC 5 n and H(c) 2 HI one has

1~- e-(‘“H)4+c,

H = H(C) .

(20)

We give a sketch of the proof of Theorem 3.19. Suppose that 6 > 0, and for some C E A we have lab _
.

(21)

We set go = [In H(ln In H)2+6] , Q = [(lnlnH)2+6] se = [In H(ln In H)3+26] ,

xo=

, c = [H@‘~~)‘+‘~]

I

f(z) = 9, $y C&“+‘b)” k=OI=0

1nlnH

5&KiZ

;

1;

,

(22) (23)

(24)

162

Chapter 3. Hilbert’s

f&Z = -jy -jy c&c

Seventh Problem

+ qSak5ablZ = g

i

(25)

k=Ol=O

cps,z = 2

2

ck,l(k

+ lb)sakzC1z

.

(26)

k=Ol=O

Using the Dirichlet pigeonhole principle and the inequality (10) of Chapter 1, one shows that there exists a nontrivial set of rational integers Ck,J, k = 07. ..,qe,l=O ,..., q,suchthat for

cp*,z --0

0 < Irk%XIck,ll 5 c .

(s,z) E Nso,Zo>,

(27)

Clearly, this first step of the proof is almost identical to the beginning of the proof of Gel’fond’s Theorem 3.2. The only difference is that, instead of a’, which we are not assuming to be algebraic, we use a number C E A that is a good approximation to ab. But now one must proceed in a different way than before. Unlike in the proof for Hilbert’s seventh problem, we do not have a large number of zeros of f(z) at integer points. Instead, we have the relation (27), which, when combined with (21)-(26), enables us to obtain the inequality f(S)

@,)

I

<

H-4(‘”

In H)4+2J/di”

I” In H

(28)

I

for (s, Z) E n(ss, ~0). We cannot use the maximum principle; however, it can be replaced by Hermite’s interpolation formula, which we will later have occasion to work with frequently and fruitfully:

N+l (C- ur)%dC z-c *

(29)

The function g(z) is regular in the closed region D bounded by the contour C, and ue, . . . , UM, and z are interior points of D. The contour C, is a circle centered at ur that lies in D, such that z and all uj with j # r lie outside of CT. Let g(z) = f(z), ut = t, M = ~0, C be the circle ICI = (lnlnH)2+6, and C, be the circle I< - TI = 0.5. If we estimate the integrals and use (28), we find that If(z)l 5 ~-~(lnlnH)4+26t/lnInInH, IzI < (1nlnH) 2+ja . Prom this, using Cauchy’s formula, we obtain the following inequality for the derivatives: f(s)(%)

I (%2)

5

~-~(InIn~)4+zd\/lnInInH,

I E fqso,~1>,

21 = (lnln H)2+a6 [

1.

(30)

55. Bounds

for Linear

Forms with

Two Logarithms

163

from (21)-(23), (25), (26), and (30) we now obtain the inequality

If we compare this inequality with the lower bound for nonzero cps,Zin (10) of Chapter 1, we see that cp8,Z - 0,

(s,z)

E Qn(so,a)

*

We are now able to obtain a stronger inequality than in (30), namely:

From this, using (29), the inequalities in (10) of Chapter 1, and (21), we arrive at the system of equations

p (0) = -g & c&l (k + lb)” = 0, k=O

s=O,l,...,

(Qo+ l)(q + 1) - 1 .

I=0

Since b is irrational, the determinant of this system is nonzero. Hence, we have a contradiction: the system of equations implies that all of the ck,l are zero, but this is not the case. The above method of proof has been applied many times by a number of authors to prove various theorems. New technical details have been developed in order to strengthen the inequalities, but the fundamental ideas have remained the same to the present day. 5.2. Refinement of the Inequalities (19) and (20) Using Gel’fond’s Second Method. By making technical improvements in his method, Gel’fond [1939] was able to prove a more precise inequality than (20):

I I

lna Inp - c > e-(l”w3+z

;

and in [1949c] (see also [Gel’fond 1960]) he proved the effective inequalities lab -
and

> e-,"3;nSC1)(n+lnH)In~+~(n+lnH+1)

I I

In Inp(Y _ c > ,-~Z(~+lnw2+c

.

In [1972] Cijsouw obtained the bound

I I lna 0-C

Be-

cnylnL)yl+lnn)-‘(l+lnL)=

>

L = L(C) .

It seemsthat the most precise bound that can be obtained by Gel’fond’s method has the form

164

Chapter 3. Hilbert’s

I I lncu ~np-C

Seventh Problem

cn2(nInn+lnH)2(1+lnn)-3

>e-

Unlike earlier papers, [Gel’fond 1949c] and later works contain bounds that explicitly take into account the degree of C. Stronger results that have been obtained using Baker’s method will be discussed in Chapter 4. 5.3. Bounds for Transcendence Measures. If we take into account the inequality (9) of the Introduction, we can use the bounds in the previous subsections to obtain similar estimates for the transcendence measures, namely:

Ip CabI I ’ e-cn2(n+ln and

-In (Y 1nP

H)

In2(n+ln

H+l)

ln-yn+l)

I( >I p

>

e-cn2(nInn+lnL)2(1+lnn)-S

,

where n = degP, H = H(P), L = L(P). We again emphasize that all of the constants in the inequalities in this section are effective. 5.4. Linear Forms with Two Logarithms. Using Lagrange’s formula, it is easy to obtain a trivial effective bound for the form

zln21-

ylnv,

z,y,u,w

E N.

Suppose that u” > vv. Then l~uZ-wV~(zlnu-ylnw) ]zlnZd - ylnv]

max eX; y In v<x<s In u

2 e-c(u~V)x,

X = max(z, y) .

This result can readily be extended to any algebraic numbers u and ‘u. Suppose that 1~~1> IVY], 21%# TJY,z,y E Z. We have IuZ -

vYl

= Iu51 le--slnu+ylnv

= WI

J

_ 11

yIIltJ--lIIlU etdt

0

5 1~~1lslnu - ylnvle (where we may, of course, assume that ]z lnzl - y lnv( 5 1). We use the inequality (10) of Chapter 1 to estimate the left side of the previous inequality. We obtain ]zIn21-

ylnv]

2 emCX,

c = c(u, v),

X = m=44, Ivl> .

(32)

The inequalities in $5.2 enable us to obtain much better bounds. A simple consequence of (31) is the bound

55. Bounds for Linear Forms with Two Logarithms

IG Ina0

- GlnP0I

QO,PO,ll,G n

=

> e

--cn*(nInn+lnL)z(l+ln7L)-3

E 4

7

CltlnaolnP0

#

JqCl/G)

degQ(Cl/L4,

165

=

L

(33)

0, .

Here c is an effective constant that depends only on In o. and In PO. 5.5. Generalizations to Non-archimedean Metrics. As early as [1935] Mahler proved a p-adic analogue of the Gel’fond-Schneider theorem. Theorem 3.21. Let CYand fi be p-adic algebraic integers, where

Then In, CY/In, ,l3 is either a rational number or a transcendental p-adic number. This theorem was then generalized to p-adic valuations. In [1940] Gel’fond proved the following quantitative result. Theorem 3.22. Suppose that p is a prime ideal of the algebraic number field K, Q, p E K, [al,, = l/?lP = 1, and the relation am - p” = 0 with m,n E Z holds only when m = n = 0. Then for any E > 0 Ord,, (a5 f BY) < ln3+E X X = m=(lzl,

Z,Y E z,

1~1)

2 X~(chB,p,e)

> 0.

If, in addition, pip with p a rational prime, ordpp = t, kp 2 ICI > t/(p - l), ord,,(o-1) = kl, andord,,(P-1) = kz, thenfor anyql,qz E Z with (ql,q2) = 1 one has

1nP I I -__

42

lno

ql

D

,

e-

ln3+c

q ,

Q=m4lqil,

lq21) L 40(lna,lnP,p,E)

> 0.

Multidimensional generalizations and refinements of Gel’fond’s inequalities have been obtained using Baker’s method. We will discuss this in Chapter 4. 5.6. Applications of Bounds on Linear Forms in Two Logarithms. These bounds have been used to answer many questions. We give some examples. 1) Let q(z) $0 be defined and completely multiplicative on Z, i.e., cp(mn) = cp(m)cp(n) for all m,n E Z. Suppose that cp(p) = 0 for all but finitely many primes p, and

166

Chapter 3. Hilbert’s

for some constant

Seventh Problem

c and any N E N. Then cp(p) # 0 for at most one prime

p; and if there is such a prime po, we must have cp(po) = eia, a E IL!. This theorem

was proved by Linnik

and Chudakov

[1950].

2) An example in Gel’fond’s book [1960] has played a very important role in transcendental number theory and the theory of Diophantine equations. Let (Y = o(l) and /3 = p (l) be linearly independent integers of a real algebraic number field lK = K(l) of degree three, and let lKc2) and lKc3) be the conjugate fields. Consider the Diophantine equation Norm(aa:

+ /3y) = fi(&)z

+ /?(j)y) = 1 .

(34)

j=l

This is a special case of Thue’s equation (38) of Chapter 1. By Theorem 1.11, it has only finitely many solutions; however, that theorem does not give an effective bound on the number of solutions. Gel’fond gave a method for finding effective bounds for the solutions of (34), by reducing this problem to estimating linear forms in the logarithms of algebraic numbers. The value of Gel’fond’s method was not that he was able to find effective bounds for solutions of (34) (which had already been done before using methods of algebraic number theory; see, for example, [Delone 19221 and [Faddeev 1934]), but rather that he had the idea of reducing Diophantine problems to bounds for linear forms in the logarithms of algebraic numbers. We now sketch his solution to the above problem. Suppose that K c2) and lKc3) are complex fields. In that case, by the Dirichlet unit theorem, the units of lK are of the form flm, where I is a fundamental unit. This means that (34) is equivalent to the system of equations

a(l)x+ pq/ = (10)m &d2 +pwy = (9 m (w(3)2

+ p(34/

=

p39

m

for some m E Z, where It21 and 1c3) are the conjugates of I = 1(l) in Kc21 and lKt3). If we eliminate x and y from (35), we obtain the following equation in m:

(

a(3)p(2)

_ &)p(3)

) (qm

+

(&p(3) +

We may suppose 10~s that ]1t21 ] = m satisfying (36) n > 0. Note that

(,qw

- a(3q3u))

(qm

+

(36)

- &)p@) >(I(3)>m=0.

that 1(l) > 1. Since ]1(2)] = ]1(3)] and 1(1)1(2)1(3) = 1, it fol]1t3) ] = (]I(‘) ])-lj2. It is now clear that the positive integers can be effectively bounded from above. Next, let m = -n, (~(~)/3(‘) - (~(‘lp(~) # 0, since if (Y(~)//?(~) = &)//z?(~), then

55. Bounds

for Linear

Forms with

Two Logarithms

167

we would have c~(~l/p(~) E R. But a complex cubic field contains no irrational real numbers; hence, a (l) /p (l) E Q, which contradicts the linear independence of a and /3. Now from (36) we obtain the inequality

where c and ~0 are effective constants. We fix In A. From (37) we obtain the following inequality for some r E Z:

In 1952, when the Russian edition of [Gel’fond 19601appeared, no nontrivial effective lower bounds were known for the absolute value of a linear form in three logarithms of algebraic numbers. Hence, Gel’fond introduced another condition: that X be a root of 1, i.e., In X = hi, t E Q. Note that in any case ]A] = 1, since the numbers (Y(~)P(~) - a(1)@(2) and CX(~)/~(‘)- CY(~)~(~)are complex conjugates. From (38) with k = 2r - t we now obtain I(2) nlnm-kS where cl justified, below by I(2)/I(3),

<e

--c1n

<_ e-c2L(nlk)

7

and cg are effective constants. The step from tin to c&n/k) is because the rational number n/k is bounded both from above and effective positive constants. We compare (33) and (39) with (~0 = PO= -1, & = n, 52 = k. We find that c2(n + Ikl) < ~3ln2(n + lkl> ,

where cs is an effective constant. This implies that we have the effective inequality /ml In+lkl
I4 + IYI L

c5

from the system of equations (35), where cs is also an effective constant. In [1960], p. 176, Gel’fond notes that, if one could obtain effective inequalities like (33) for linear forms in an arbitrary number of logarithms, then “we would obtain a computable bound for the magnitudes of the solutions of equation (34) in an arbitrary algebraic field K,” i.e., we would have an effectivization of Thue’s theorem. On the next page Gel’fond writes: “Nontrivial lower bounds for linear forms, with integral coefficients, of an arbitrary number of logarithms of algebraic numbers, obtained effectively by methods of the theory of transcendental numbers, will be of extraordinarily great significance in the solution of very difficult problems of modern number theory.” Later

168

Chapter 3. Hilbert’s

Seventh Problem

events proved that Gel’fond was correct. Fourteen years after he wrote those words, the program that Gel’fond was proposing was put into effect. We shall discuss this in detail in Chapter 4. 3) Using (27) and Theorem

3.22, Gel’fond

[1940] was able to prove

Theorem 3.23. Let cr, p, and y be real algebraic numbers not equal to 0, 1, Suppose that at least one of them is not an algebraic unit. Then the equation a” + p = y= or -1.

has only finitely many solutions x, y, z E Z, except in the case when a = f2a,

p = f2b,

y = f2C,

a, b, c E Q .

4) Using Theorem 3.22, Cassels [1960] obtained an effective upper bound for the solutions to the equation X-Y=M,

MEN,

in natural numbers X and Y all of whose prime divisors belong to a fixed finite set. In $4 of Chapter 1 we saw how to obtain a non-effective version of this result. 5) Suppose that m E N, (a + /?)2 and crfi are rational integers, (Y # fi, and p, =

(cP - /P)/(cY -/I), { (a” - fP)/(cr2 - p”),

if n = 2m + 1, if n = 2m.

The P, are called Lehmer numbers. Obviously P, E N. A prime p is said to be a proper prime divisor of P,, if p/P,, and p ]Pk for 0 < k < n. It was known that if o and p are real and if (a@, (a + ,B)2) = 1, n # 6, n # 12, then the numbers P,, have proper prime divisors. In [1962] Schinzel used Gel’fond’s inequalities for the form x In a-y In /3 to prove that, if (Yand /? are complex and if a/P is not a root of 1, then the numbers P, have proper prime divisors for all n >_nc(oL, 0). This implies that the largest prime divisor of P, approaches infinity as n increases. Later, Schinzel [1967] found an explicit formula for no. It is clear from the definition of the Lehmer numbers that (Y and p are algebraic of degree 1, 2, or 4. Subsequently the numbers P,, were generalized using arbitrary algebraic numbers (Y and p, and a proper prime divisor was interpreted as a prime ideal. 6) In [1967] Schinzel refined Gel’fond’s inequalities and obtained explicit formulas for the constants. Here are some results that follow from his estimates. Theorem 3.24. Suppose that uo, ~1, P, Q E Z, PQ # 0, A = P2 - 4Q < 0, P2 # jQ for j = 1,2,3, uf - Puluo + Qui # 0, and un+l = Pu, - Qu+l for n >_2. If q is any prime divisor of Q/(P2, Q) and n > no, then

$5. Bounds for Linear Forms with Two Logarithms

lunl > &(pZ.Q)“/‘exp Here no can be expressed explicitly Schinzel also obtained of u*.

(iv-)

169

.

in terms of P, Q, q, uo, ~1.

an effective lower bound for the largest prime divisor

Theorem 3.25. Let D be an odd natural number not equal to 2” - 1 for k E N. Then 1) The Diophantine equation

x~+D=~~ has at most one solution with m > 80 and x > 0. 2) If p is any prime divisor of 40 + 1, then the Diophantine equation x2+D=pm has at most one solution with m > 1 + 6(10 + lnlnp)/ Theorem

3.26.

lnp .

Let CYbe a real quadratic irrationality,

and let g E N, g > 1.

Then

Il~dYl > c7+ exp(43 for n E N, where c is an effective constant.

For a E Z let P(a) denote the largest prime divisor of a. Theorem

3.27.

Suppose that f(x)

E Z[x], deg f (x) = 2, and f(x)

has sim-

ple roots. Then

wherek=2iff(

lim inf p(f (x>> > ~ z-boo lnlnx - 7 ’ x ). as irreducible and k = 1 otherwise.

7) Let R = Q(A) be a quadratic imaginary field, where -d E N is squarefree. It is well known that any cr E ZK can be uniquely written in the form a+bw,a,bEZ,where 4-i w = { i(l+

A),

ifdE2,3(mod4), if d E 1 (mod 4) .

If w’ is the conjugate of w (i.e., either -&

or (1 - &)/2),

DC1 w2 I I 1 w’

= (w - w’)2

is called the discriminant of K. It is easy to see that

then

Chapter 3. Hilbert’s

170

D =

Seventh Problem

4d,

if df

2,3 (mod 4),

d,

ifdrl

(mod4).

Unlike Q, most quadratic imaginary fields do not have unique factorization into prime elements. One first has to decide what numbers play the role of primes in algebraic number fields. Let a,P E Zx. We say that cr and p are associates, and write cr - /3, if (Y = ,0(‘ f or some algebraic unit C. If p E & is not a unit, we say that p is irreducible in # if all elements of Zn< that divide p are either associates of p or units. In Q there are only the two units fl, so that k - m for k, m E Z simply means that lc = fm. The irreducible elements of Z are of the form p or -p, where p is a prime. We say that the field lK has unique factorization if, whenever 7rl

. * .7r,

=

p1

* * * pt

with irreducible ni and pj, we have s = t, and each ni can be paired with a in corresponding pi so that pi - pj. Besides Q, we have unique factorization such fields as Q(G) and Q(n). On the other hand, lK = Q(a) does not have unique factorization, as we see from the example 2.3=(1+-)(1-G), in which all four numbers are irreducible, and 2 and 3 are not associates of l&G. For a long time nine values of d were known for which the corresponding quadratic imaginary field has unique factorization, namely: -1, -2, -3, -7,-11, The corresponding

discriminants -4,-8,

-19, -43,-67,

-163.

(46)

-67, -163.

(41)

are

-3, -7,-l&

-19,-43,

It was Gauss who found these discriminants, although question that at first looks completely different. Let A,B,C

E Z,

D=B2-4AC
he was studying

a

A>O,

F(x, y) = Ax2 + Bxy + Cy2 . F(z, y) is called a positive definite binary quadratic form, and D = D(F) is called its discriminant. If @(x, y) is another binary quadratic form, we say that F(z, y) and G(x, y) are equivalent, and write F - I, if there exist a, b, c, d E Z, ad - bc = 1, such that @(x, y) = F(az

+ by, cz + dy) .

(We note that equivalence of quadratic forms can be defined in much greater generality; but we shall only need to consider positive definite binary quadratic forms.)

$5. Bounds for Linear Forms with Two Logarithms

171

It is easy to check that D(F) = D(Q) for equivalent forms F(s, y) and @(z, y). Since F - @ and F - G imply that @ N G, we can divide all forms with a given discriminant D into equivalence classes. It can be shown that for fixed D there are only finitely many equivalence classes. Gauss wrote that the nine numbers (41) are all the discriminants he was able to find that have only one equivalence class of forms. These are called the discriminants of class number one. A basic concept of algebraic number theory is that of an ideal class in a field lK c A. We say that two ideals 2t and 93 of K belong to the same class if there exist a,@ E Zx, (YP # 0, such that (o)!2l = (/?)%, where (o) and (fi) denote the principal ideals generated by (Y and ,0. It turns out that K has unique factorization if and only if it has only one ideal class, i.e., if and only if all ideals of the field are principal. Now we need only observe that there is a one-to-one correspondence between ideal classes of a quadratic imaginary field of discriminant D and equivalence classes of positive definite binary quadratic forms of the same discriminant D. Thus, Gauss’ nine numbers (41) are also the discriminants of quadratic imaginary fields with unique factorization. Gauss conjectured that there are at most finitely many discriminants miss.ing from his list of discriminants of class number one. This conjecture attracted the attention of many mathematicians. Finally, in [1934] Heilbronn and Linfoot proved that there can be no more than one discriminant missing from the list (41), and that such a tenth discriminant DIO, if it exists, must be large. Increasingly large lower bounds were found for DUO: Dickson [1911] obtained IDlo > 1.5. 106; Lehmer [1933] proved that lDl,-,I > 5.10’; and Stark [1967] was able to show that ~DIO[ > exp(2.2. 107). In [1948] Gel’fond and Linnik proved that, if DIO exists, then the following inequality holds: ]zilnor

+22lncYs

+lncrs]

<e --Y&m.

Here ~1, (~2, and (~3 are fixed algebraic numbers independent over Q; x1,22 E %, with l~ll,l~2l

whose logarithms

(42) are linearly

< -Yom;

and y and 70 are effective constants. In 1948 no one had yet been able to prove a nontrivial and effective lower bound for the absolute value of a linear form with three logarithms of algebraic numbers; hence, (42) had to be kept in reserve. If it became possible to show that the absolute value of the linear form in (42) is bounded below by exp(-cp(]Dru])), where cp(z)/& + 0 as x + 00, then one would have an upper bound for [DIo~. It would then just be a matter of comparing this upper bound with the lower bounds given above, and, if they are incompatible, then we would know that DIO does not exist. This plan came to fruition twenty years later. We shall return to this question in Chapter 4.

172

Chapter

3. Hilbert’s

Seventh Problem

In [1952] Heegner published a proof that Die does not exist. But his proof contained a gap, and he had to withdraw his claim. The deficiency in Heegner’s proof was finally corrected by Stark in [1967, 19691.But actually the solution had been within reach of Gel’fond and Linnik. As Chudakov showed in [1969], the problem of showing that Dia does not exist can be reduced to finding a suitable bound for a linear form with two logarithms (this was also shown independently by Stark). Chudakov proved the inequality IHln(2 + &)

- 2?rmI

< 73.5e-“ml12,

HEZ.

(43)

After making the computations, it turned out that J&e] < 1042, and this bound is incompatible with Stark’s inequality l&e] > exp(2.2 . 107). 8) Gel’fond’s inequalities were used in [Babaev 1966] and [Segal 1939, 19401 to answer questions about the distribution of integer points on certain curves and about the distribution of integers that are divisible only by primes in a fixed finite set. 9) A final application is connected with coding theory. In [Bassalygo, Fel’dman, Leont’ev, Zinov’ev 19751the inequality (zln3 - yln2]

> e-210(‘nY)2,

Z,Y E N,

Y22,

is used to solve a problem related to error-correcting codes.

96. Generalization of Hilbert’s Seventh Problem to Liouville Numbers 6.1. Ricci’s Theorem. In [1935] Ricci showed that the number a in Theorem 3.1 can be replaced by any Liouville number. Theorem

3.28.

Let a,/? E A, (Y # 0, 1, p $! Q. If K. $ Q, and for some E > 0

the inequality K, I

has infinitely scendental.

!2

<

e-(‘“q)

S+e

9 I

many solutions p E Z, q E N, then the number (an)0 is tran-

Theorem 3.29. Let a,/3 E A, ,f3 4 Q, n $ Q, QK # 0,l. some E > 0 the inequality

n-P<e 9I I

-(ln

Suppose that for

q)a+c

has infinitely many solutions p E Z, q E N. Then the number (cr~)fl is transcendental.

$7. Numbers

Connected

with

the Exponential

Function

173

6.2. Later Results. In [1937] Franklin proved Theorem 3.30. Let IK be a fixed algebraic number field of finite degree. Suppose that a # O,l, b 4 Q, and

where &, &,6,, E IK are irrational and have conjugates that are uniformly bounded. Further suppose that A, E N and &A,, &A,, &A, E 23~. If la - EnI + lb - &I + lab - &I < e-Inb An for n = 1,2,. . ., then k 5 7.

In [1957] Schneider proved that the inequality la - aI + lb - /3[ + la” - yI < e-Ins+’ H,

&>O,

has only finitely many solutions (Y,p, y E A, p $ Q, H = max(a(a), H(P), H(y)), if the degrees are bounded. Schneider’s inequality was later refined by a series of authors, including Bundschuh, Cijsouw, Mignotte, Shmelev, Waldschmidt, and Wiistholz. We give a result from [Waldschmidt 1978b]. Theorem 3.31. Let a, b E Cc, aln a # 0. Then for any (Y,/3, y E A, ,6 $! Q, one has Ia - aI + lb - @I+ lab - yI > e--cn4(‘nH)3(‘n’nH)-2 , where H = max(H(a), constant.

H(P), H(y))

and c-c-

(1na,blna)>Oisane#ective

The condition that /I $?!Q cannot be removed, as one seesfrom the following theorem of Bijlsma [1977, 19781. Theorem 3.32. Given any K.> 0, there exist irrational such that the inequality la - al + lb - PI + lab - yl < e-In” H,

numbers a, b E (0,l)

H = m=(H(a),

H(P), H(y))

,

has infinitely many solutions (~,p,y E Q.

$7. Transcendence

Measure of Some Other Numbers Connected with the Exponential Function

7.1. Logarithms of Algebraic Numbers. In Chapter 2 we discussedthe irrationality measures and the transcendence of 7rand In Q for Q E A. The bounds in that chapter were obtained by methods that various authors developed using ideas of Hermite. We had good bounds in terms of H for ]e -
174

Chapter 3. Hilbert’s

Seventh Problem

height H. Sometimes these bounds were nearly optimal as a function of H (for example, Mahler’s inequality (69) in Chapter 2). However, as a function of n they were usually far from optimal (see (6) of the Introduction). Gel’fond’s second method enables one to prove bounds on transcendence measures that are much closer to optimal in n as well as H. Many mathematicians have used Gel’fond’s method to derive various bounds for the transcendence measures of rr and lncr - Waldschmidt, Diaz, Cijsouw, and others. We shall give several examples. We begin with bounds for approximations by algebraic numbers. Theorem

For any C E A, deg< = n, L(C) = L, one has

3.33.

I7r - Cl > e- cn(nln

n+lnL)(l+ln

7%)

7

(44)

where c is an effective constant. If we compare (44) with (7) and (9) of the Introduction, we see that for In L >> n Inn the only factor that might be “unnecessary” in the exponent of (44) is the increasing term (1 + Inn). The constant c has been computed; in any case, it is known that one can take c = 389. We note that stronger inequalities can be obtained under some additional conditions on C, for example, if C E Q(e2AilN), iV E N. Theorem 3.34. Szlppose that m 2 1, cri, . . . , Q, E A, and lnoi, . . . , lna, are fixed valzles of the logan’thm that are linearly independent over Q. There exists an eflective constant c = C(CQ,. . . , cr,; In cyl, . . . , In (Ye) > 0 such that for any
1 lnak

_

CkI

>

e-cn’+“~(nlnn+lnL)/(lnn+l)

)

k=l

where

nlnLi -+...+m

L = exp nk

=

d%
Lk

n = deglK,

=

nlnL,

> > IC=l,...,m,

nm L(
~=~((yl,...,~Y,,~l,...,~,).

By comparing (45) with (7) and (9) of the Introduction, we see that the factor n’l”/(lnn + 1) is the only part of the exponent that might not be necessary. This theorem can be improved in the casewhen (~1 = -1, i.e., when In cri = (2t + l)si, t E Z. (One could take a instead of xi, but in that case lK has to contain i.) One can obtain the inequality m In

-

cl 1 +

C k=2

1 ln

ak

-

ck 1 2

eTcntnln

n+ln

L)(ln

n+l)l’m

.

57. Numbers Connected with the Exponential

Function

175

When m = 1 this is the same inequality as (44). The inequalities (44)-(45) with m = 1, together with (8) of the Introduction, enable one to obtain lower bounds for the transcendence measures of x and lncu for (Y E A. Theorem 3.34 makes it possible to construct more examples of transcendental numbers. If a number p has approximations by algebraic numbers that satisfy inequalities incompatible with (45) for m = 1, then p cannot be the logarithm of an algebraic number, and so expp is transcendental. There are many ways of constructing such p: as series, infinite products, continued fractions, and so on. For example, one could take

where nP = 4P2 for p E N. 7.2.

Approximation

of

Roots

of

Certain

Transcendental

Equations.

Gel’fond’s method enables one to estimate the transcendence measures of numbers that satisfy certain transcendental equations, for example: P(z,a”)

= 0

or

P(z,e”)

= 0,

where a E A, P(x, y) E Z[z, y]. H ere is a result from [Shmelev 19701. Theorem 3.35. Suppose that P(x, y) E Z[x, y], PLPi f 0, and P(x, y) has no nonconstant divisors in Z[x]. Suppose that p $ Q satisfies the equation

P(z,a”)

= 0,

a E A,

ulna # 0.

Then there exists an effective constant c = c(a,n) > 0 such that

IP- Cl> e-c(ln

fIq(ln

In Hy

for any C E A, deg< 5 n, H(6) 5 H. Corollary 3.2. Let p 4 A, n E N, a E A, aln a # 0. Suppose that for any CQ> 0 the inequality -co(lnH)s(lnInH)”

IP - Cl < e

has infinitely many solutions C with degc 5 n, H(c) 5 H. Then p and up are algebraically independent.

176

Chapter 3. Hilbert’s

$8. Transcendence

Seventh Problem

Measure of Numbers Elliptic Functions

Connected

with

In 53 we gave several theorems on transcendence of numbers connected with the Weierstrass function Q(Z). Bounds for the transcendence measure of such numbers have been obtained in many papers by such authors as Beukers, Brownawell, Chudnovsky, Hirata-Kohno, Kholyavka, Masser, Nagaev, Philippon, Reyssat, Tubbs, and Waldschmidt. We shall give some of these inequalities below. We shall use the notation of $3 and the notion of a multiplier of a lattice that was introduced in $3.2. 8.1. The Case of Algebraic Invariants. In this subsection we shall suppose that the invariants g2 and g3 of p(z) are algebraic. We further suppose that t,tL,Ez E A, n = degt, nj = degtj, H = H(C), Hj = H(&), L = L(C), Lj = L(&), j = 1,2. All of the c and cj will be effective positive constants. Here are some of the many inequalities that have been proved:

1) ]W - 51 > exp (-c(w)n2(n(lnn)3

+ (InHlnlnH)‘)),

2) ]wi - &I + ]WZ- 521> exp(-cvl\rln’iV),

w E R.

where

3) If & wi + &WZ # 0, then ISlwl +

Gw2l

>

exp(-c2vM3)

,

where M=lnN+vlnv+vlnln(N+l)+civ,

. 4) Let u be an algebraic point of p(z), i.e., p(u) E A. Then ]u-El

> exp(-csn4(lnH(lnlnH)2+n(lnn)3)(1+lnn)-4)

.

5) Withuasin4),

147~)- El > exp (- c4n4(ln H(lnln 6)

Iv - 51> exp (- csn2(In H(ln

H)’ + n3(lnn)‘)(1

+ Inn)-4)

In H)’ + n2 (ln n)“)) .

These inequalities are essentially due to Reyssat [198Oc].

.

58. Numbers

Connected

with

Elliptic

Functions

177

In [1990a, 1990b] Hirata-Kohno obtained a refinement of 3) and 4) and also a new inequality: 3’) ]&wi + &wz] > exp (-csv3(lnN 4’) Iu - (1 > exp (- crn2(lnH 7) I&w +

(274

+ lnv)(lnlnN

+ nlnn)(lnlnH

+ lnv)2).

+ lnn)2).

2 exp (- cs~2(lnN+lnv)(lnlniV+lnv)).

Because of the inequality (8) of the Introduction, one can obtain similar bounds for the transcendence measures of these numbers. 8.2. The Case of Algebraic Periods. From $3 we know that both wi and ws are transcendental whenever gs, gs E A. Hence, if wi, ws E A, then at least one of the numbers gs,gs is transcendental. Notice, by the way, that in that case one of the invariants might still be algebraic. For instance, if the lattice is constructed from the basis { 1, i}, then gs = 0 E A. We give two theorem of Kholyavka [1987].

E A, n = degQ(wi,ws,&,&),

Theorem 3.36. Suppose that wl,w2,<2,{3 nj = degJj, Lj = L(&), j = 2,3, and N = (122 + n3)

&lnLs

+ zlnLs+n

(

)

+nlnn.

Then 192-t2I

+ 193- 631> exp(-cnNlnN)

Theorem 3.37. Suppose that &,<s,wi Ns = (ns +ns)

(

.

E A, n = degQ(&,&,wi),

and

2

elnL2

+ 5lnLs

+n+nlnn

>

.

Then

I92-

(21 + I93 - 531 > exp (-cn2NslnNo)

.

8.3. Values of p(z) at Non-algebraic Points. Bijlsma [1977] extended the results of s6.1 to p(z). Theorem 3.38. Let g(z) have algebraic invariants and period lattice R, let lK be the field of multipliers of p(z), and let a, b E Cc,a, ab $! 0. Then for any a,p, y E A with /3 4 K one has ]@(a) - (Y)+ l&ab) - PI + lb - y] > exp (-cn6(lnH)6(lnlnH)-5)

,

where n = deg Q(Q, P, r>,

H = madee, H(a), H(P), H(y))

Here c is an effective constant depending only on p(z).

.

178

Chapter 3. Hilbert’s

Seventh Problem

Bijlsma also extended his Theorem 3.32 to p(z), and proved that the condition p 4 K cannot be removed in Theorem 3.38. In [Bundschuh 19711 bounds were found for the transcendence measure of roots of an equation of the form P(z, p(z)) = 0, where g(z) is the Weierstrass function.

51. Linear

Forms in the Logarithms

of Algebraic

Chapter 4 Multidimensional Generalization Seventh Problem $1. Linear

Forms in the Logarithms

Numbers

179

of Hilbert’s

of Algebraic

Numbers

1.1. Preliminary Remarks. Let (~1,. . . , cr, E A, and let In ai,. . . , Ina, be fixed values of their logarithms. For the duration of this chapter we will use the notation A=bilnai+~~~+b,lna,,

bi,...,b,EZ;

Ao=bo+bllnalf...+b,lncr,, Al =bilnai

b0,

+*..+b,lna,,

(1)

h ,...,b,

bi,...,&

E A;

(2)

EA.

(3)

It is easy to prove a trivial lower bound for IAl. Let x+1...&-

0 < IAl < 1,

-1.

If L is the line segment connecting 0 and A in the complex plane, then

1x1= /”0 I

e’dz

5 IAl

rnLp leZl < e[Al .

I

The number X is the value at (crYf’, . . . , a%), &j = sgn bj, of the polynomial

Since A # 0, it follows that X = 0 is possible only when A = 2kri for nonzero k E Z, in which case \A( >_ 2n. So we may assume that X # 0. From (11) of Chapter 1 we obtain the inequality IAl 2 emCX,

x = ozym -- lbjl 3

where c = c(cri,..., a,) > 0 is an effective constant. (In the case m = 2 a similar estimate was obtained in 55.4 of Chapter 3.) The inequality (4) can be strengthened using any of the ThueRoth theorems. This is what Gel’fond did in [1948]. Theorem 4.1. Let E > 0. There exists a constant He = He(c,ai,. lncri,..., lna,) > 0 such that if A # 0, then IAl > emEx,

X = oFjym -- lbjl 2

HO

.

. . ,cY,,

(5)

The constant He is non-effective, since (5) comes from the non-effective Theorem 1.16. Gel’fond also obtained a padic analogue of Theorem 4.1.

180

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

In the case m = 2, Gel’fond’s method was used to obtain bounds that

are stronger than (4) or (5), are effective,and apply to forms with algebraic coefficients. This was discussed in @5.4-5.5 of Chapter 3. 1.2. The

First

Effective

Theorems

in the General

Case. In [1966a]

Baker

proved the following fundamental result. Theorem 4.2. Suppose that m 2 2, cr1, . . . , cxy, E A, 6 > m + 1, n E N, and the numbers In (~1,. . . , In cr,, and rri are linearly independent over Q. There exists an eflective constant c = c(m, al,. . . , am, In (wl, . . . , lna,, K, n) > 0 such that for any nontrivial m-tuple PI,. . . ,p,,, E A, deg/?j 5 n, H(fij) 5 H, j = l,..., m, one has IAl 1> ce-lnffi H . (6)

When m = 2 this inequality is the same as (27) of Chapter 3. Baker gave two consequencesof his theorem that resolve the multidimensional analogue of Hilbert’s seventh problem. Theorem 4.3. Suppose that ~1,. . . , cx,,, are real, algebraic, and not equal to 0 or 1; and supposethat PI, . . . , &,, are also real and algebraic. If 1, pi,. . . , & are linearly independent over Q, then the number

is transcendental. Theorem4.4. Letcq ,..., cr, l A,crl...cr, #O. Ifthenumberslncxl,..., and rri are linearly independent over Q, then they are linearly indepenlnff,, dent over A.

When m = 1, Theorem 4.3 is the same as Theorem 3.3, and Theorem 4.4 is the same as Theorem 3.1, except that the conditions in Theorems 3.1 and 3.3 are less stringent. Soon after, Baker [1967b, 1967b] refined his theorems as follows. Theorem

4.5. Suppose that /3&l * * *,&

H = mm H(pj),

# 0, IE > m + 1, n = maxdeg&,

j = 0, 1, . . . ,m. Then IA01

wherec=c(m,cul,...,

o,,lncri

>

ce

-In=

H

>

,... , In a,, n, n) > 0 is an effective constant.

Theorem 4.6. Suppose that either In ~1,. . . , lncw, or else PI,. . . , & are linearly independent over Q. Let n = maxdeg& and H = max H(/?j). If K > m, then IAll > ce-I”” H ,

wherec=c(m,ai

,...,

(r,,lnal,...

, In o,, 6, n) > 0 is an effective constant.

31. Linear

Forms in the Logarithms

of Algebraic

Numbers

181

Theorem 4.7. Let (~1,. . . ,CX, E A, cq *..cr, # 0. Then lnoi ,..., lno, are linearly independent over A if and only if they are linearly independent over Q.

Ifcq

Theorem4.8. number

,...,

(wm,/30,P1,...,

,&EA,~~~~~%&#O,

thenthe

eoaap . . ..om m

is transcendental. At the 1970 International Congress of Mathematicians in Nice, Alan Baker was awarded the Fields Medal for his series of papers on bounds for linear forms in the logarithms of algebraic numbers and their applications to various problems of number theory. Theorem 4.5 also has consequences for the theory of integrals of rational functions. In his book [1949] Siegel noted that transcendental number theory had not at that time been able to determine the algebraic nature of the number

J

’ dx -= 0 2s+1

?ln2+A. 3 3&

Theorem 4.5 implies that this number is transcendental; and, in fact, one can prove a much more general fact (see [van der Poorten 19711). Let P(z), Q(z) E a[~], (P(z), Q(z)) = 1. Let ~1,. . . ,cr, be all of the distinct poles of the function P(z)/&(z), and let vi,. . . , vm be the corresponding residues. If the contour r in Ccis a closed path, or if it joins two points or extends to infinity, and if the integral

Jrp(“)dx Q(z) m SC

exists and is equal to a, then a E A if and only if

rkcl

“kdz=O. z--k

In particular, if deg P(z) < deg Q(z), then either a = 0 or else a $ A. In 1968 the exponent K in (7) was replaced by 1. Theorem 4.9. Let In al,. . . , lno, be any fixed values of the logarithms of the algebraic numbers al,. . . , crm, and let n E N. There exists an effective constant cs = cc,(ai ,..., om,lncri ,..., lna,,n) > 0 such that if -40 # 0,

deg~(al,...,cr,,po,...,P,)


then lAol > L-co,

L

=

o~k~mL(Pk) --

.

(8)

When m = 2 and PO = 0, this theorem gives a better bound in terms of L than (29) and (32) of Chapter 3. On the other hand, the theorem does not

182

Chapter 4. Generalization

show how CXJdepends on 3. Theorem 1.9 shows us Lmco cannot be replaced power of L. The exponent because of its importance detail.

of Hilbert’s

n, whereas

Seventh Problem

this is done in (29) and (32) of Chapter

that the dependence on L in (8) has the right form: by a function that decreases more slowly than a c~ has been the subject of much research, especially in applications. In $1.4 we shall discuss this in more

1.3. Baker’s Method. Many authors have introduced various technical refinements - some of which are rather complicated - in order to strengthen Baker’s original bound (6). However, Baker’s basic idea that allowed him to improve on Gel’fond’s second method still lies at the heart of the proof of these results. We shall describe this basic idea without dwelling on the technical details, which can be found, for example, in [Baker 1975a], [Stolarsky 19741, [Waldschmidt 1974c, 1979b]. For simplicity, we shall assume that cq, . . . , cr,, pi,. . . , ,&-i E A, and shall sketch the proof of the following assertion: if

lna,

=/3ilncrl

+...+/3,-ilnom-r,

(9)

then lnai,... , In CX, are linearly dependent over Q. (Note that when m = 2 this is Theorem 3.3.) 1) Let q E N, n = degQ(m,. s = (4mq)2m+l, LqU,V)

={(s,z)

. . ,a,,

c = 2qsm,

x =qm, : Sl)...,

PI,. . . ,&-I),

Sm-l,ZENO,

s= (Sl)...)

sl+“‘+Sm-l

3,-l)


XIV}

) (10) )’

j=l

kl,...,k,=O

Then

= 5 ... E c,E’(k, + ~mPj)sj(lnaj)sjOI~~j+k,Pj)z , kl=O Hz,.

j=l

k,=O . .

, z) = @s(z) ln” cri . . . Ins”-’ cr,-1 .

The functions @s(z) can also be obtained in another way. If m-1 cp(z)

=

cc,

rl;

k

j=l

q+ojk,)=

)

51. Linear Forms in the Logarithms

of Algebraic Numbers

183

then for s E N

p(z) =

c S!k’ q%qz) sl+...+s,=s j=l

.

3’

In what follows it makes no difference whether we regard the e(z) as coming from f(y) or from v(z); in fact, neither f(y) nor P(Z) will appear again. In view of (9), the function e(z) can be written in two different ways: e(z)

=

C

(7,

m-l JJ

((5

j=l

.

+

pjkm)sjajkj+Bjkm)r) I

h

(11)

m-l =

(kj

c,

fi

+/?jkm)'j

cn j=l

=k

N

a:’

.

j=l

1

Using Theorem 1.9, the inequality (11) of Chapter 1, and the relation can prove that there exist C, E Z with 0 < max IC’zl 5 C such that c&(z) These conditions

= 0

for

(a, 2) E w,

X)7

can be realized because the cardinality 42m3+m2

while the number of coefficients

9

2m2-1

9 2 90 -

(9), one

(12)

of a(,!?, X) is at most

7

Cz is equal to (q2” + l)“,

2) From (11) we obtain

@P’(z) =tl+...+t,-,=t c t!

m-1 lnt’ cr, j--J t%+S(z) r=l 7-S

8

9

(13)

where? = (tl,.. . , tmel) E Nr-‘. Th’ 1s equality plays a central role in the method, since it allows one to express the derivatives of the functions @z(z) in terms of the same functions. We can now use induction. Let T,Y

E N,

X

2T > S . 16-m2 .

If we suppose that @&)

for

= 0

(a,%) E fWT,Y)

,

(14)

then from (13) we find that &‘(z) 8

= 0

for

t
(S,Z)ER(T,Y).

Thus, for sr +. . .+sm-i 5 T the functions e(z) have a large number of zeros of high order. Using the maximum principle, it is then easy to prove that

184

Chapter

4. Generalization

max I&(z)1 5 ~-y~q’~+l, MlYJTi

of Hilbert’s

Seventh Problem

y > 0, s1 +. . . + s,M1 5 T .

For z E Z we compare (15) with the lower bounds for e(z) (11) of Chapter 1. We see that @a(x) = 0

for

(S,x)~fW,Yd3,

(15)

that follow from

qLm2qo.

(16)

In other words, (12) can be extended to a larger set of values of z (although the range for s has to be reduced by a half). If Y = X and 2T = S, then (14) holds, and we can start the induction. After 4m2 - m steps we obtain the inequality

;z;z If(z)1 = pg PO(Z)1L q-7q2m2+2m+1 . Using Cauchy’s formula for the derivatives, we then find that @p(o) 5 q-P2+2m , I I

s = 0,. . . , Q,

Q L Q2 2 Ql 7

(17)

where Q = (q2” + 1)” - 1. We consider the following system of equations in the variables Cz: @t’(O) =CCz(krlncYr :

s=O,...,Q.

+...+IC,lna,)‘,

(18)

If all of the numbers Icr In (or+ .. . + &a,,, are distinct, then the determinant A of the system (18) is nonzero. If we estimate the ratio of the minors of A to the determinant A, from (17) we find that

for all CF. But C, E Z, and so all C, = 0, contradicting (12). This means that A = 0, i.e., for two different vectors x’,%” E Z” we have Icilnar

+...+Ic~lna,

=!i$lnar

+...+kklna,.

Thus, the numbers lncrr, . . . , lncr, are linearly dependent over Q. In his first paper [1966a], at the last step Baker used the following system of equations in CT: v(x) = 0,

x = l)...

) (q2” + 1)” .

It has a nontrivial solution; hence, the determinant of the system is zero. This means that some of the numbers a:, . . . (Y> must be equal. In order to obtain a contradiction from this, in Theorem 4.2 he introduced the condition that the numbers In (~1,. . . , In am, and 2& be linearly independent over Q. The main way in which Baker’s method differs from Gel’fond’s is that he divides up the derivatives of p(z) into terms of the form

$1. Linear Forms in the Logarithms

of Algebraic Numbers

185

m%(z) , where the numbers Bz depend only on s and In (~1, . . . , In (~~-1; and he carries out the interpolation for each function e(z) separately. The relation (13) enables him to realize this plan. 1.4. Estimates for the Constant in (8). Many papers have been devoted to proving estimates for the constant in (8). Such mathematicians as Beukers, Loxton, Matveev, Mignotte, Philippon, van der Poorten, Shorey, Stark, WaldSchmidt, and Wiistholz have worked on this question. We shall only give a few of their results. We begin with an inequality from [Baker 19721 that depends upon the maximum length of the oj. Theorem 4.10. Let A = maxi<j<mL((Yj) = L(a,), and let n = -= max deg aj. There exists an effective constant c depending only on m, n, (~1, “‘7 (~~-1 and the choice of branches of the logarithms of aj, j = 1,. . . , m, such that if A # 0, then

In [1973b] Baker cients.

extended

this inequality

to forms

with

algebraic

coeffi-

Theorem 4.11. Let degai, degfij 5 n, 1 5 i,j 5 m, A = L((Y,) = mal<j<,,, L(aj). There exists an effective constant c depending only on -(~~-1 and the choice of branches of the logarithms, such that m,n,m,..., if A, # 0, then ]Ar] > (HlnA)-“InA,

H = lcym --

H(Pj)

(20)

.

In [1976] Shorey published a bound that takes into account not just the parameter of maximum length.

all of the

m,...,%n,

Theorem 4.12. Suppose that H(cri) 5 Ai, where Ai 1 27, and lnai is the principal value of the logarithm, i = 1, . . . , m. Further suppose that H(Pj) 5 H for j = 1,. . . ,m, where H 2276, degQ(ar ,..., (r,,fii ,..., ,&) =n, 0 = 1nAi ***InA,,

E=lnQ+lnlnH.

For any E > 0 there exists c = C(E) > 0 such that if A, # 0, then ]Ai] > exp (-(mn)CmR(ln In [1977] Baker found stronger

n1n(OH))2E2”+2+‘)

.

(21)

bounds.

Theorem 4.13. Suppose that lnaj is the principal value of the logarithm of the algebraic number “jr H(aj) 5 Aj, and H(Pj) 2 H, where H, Aj > 4, j=l ,,.., m. LetdegQ(ai ,..., am,/30 ,..., pm) =n,

186

Chapter 4. Generalization

of Hilbert’s

0 = 1nAr ***InA,,

Seventh Problem

0’ = R/lnA,

.

If AIJ # 0, then

lAoI > (Iuy”‘““‘,

c = (16r71n)~~~”

.

(22)

.Theorem 4.14. Suppose that the conditions in Theorem 4.13 are filfilled, and in addition PO,. . . , & E Z. If A, # 0, then

IAll > IrCol”“‘,

c = (16rnr~)~~~” .

(23)

Of course, in the definition of R’ we are supposing that (Y,,, is the oj of maximum height. Thanks to an important technical improvement in the proof, it was later possible to remove the factor In R’ from the exponent in (22) and (23). Theor.em4.15. Let Aj = max(H(aj),exp]lnaj],em), j = l,..., max(Ar,... ,A,,ee),H=maxl<jlmH(Pj);deg~(al,...,cr,,~oOo,...,Pm)~ n; andn=lnAr *-*InA,,,. IfAs # 0, then lAoI

2 e-

cl L?(ln

H+ln

In A)

,

c1

28m+53m2mnm+2.

=

m; A =

(24

In addition, if a1bl “‘Cl, bmfl,

H 2 m=4hl,. . . , lb4, e>,

then IAl 2 lFzR

,

(25)

where c2 = c2(m, n) is an efiectiwe constant. Theorem 4.16. There exists an effective constant c3 = cg(m,n) > 0 such that if A, # 0, then IAll > (IYK?)-~~~ . (26)

The bounds (24) and (25) were proved in [Philippon, Waldschmidt 19881, and (26) was proved in [Wiistholz 1987a, 1987b, 19881.More precise values for the constants were obtained in [Waldschmidt 1991a, 1991b]. One of the results in [Baker, Wiistholz 19931was that, if n = degQ(ar, . . . , am), lnq denotes the principal value of the logarithm, A # 0, H = max(]br], . . . , ]&],e), and Aj = max(H(q),e) for 15 j < m, then one has the bound In IAl 2 - (16rnr1)~(~+~) In Al . . . In A, In H . The bounds for A, A,, and A, can rather easily be transformed into bounds for the corresponding exponential expressions. Of course, if eZ - 1 is small it does not necessarily follow that z is small; however, one of the values of the logarithm of e’ must have absolute value close to let - I]. Lemma

4.1.

If

Iz( 5 0.5, then

51. Linear Forms in the Logarithms

of Algebraic Numbers

187

The left inequality actually holds for any z E c. Now suppose that z is one of the numbers A, Ai, As. Using Lemma 4.1 and the above bounds on these numbers, we can immediately obtain lower bounds for the absolute value of @ . . . cr> - 1,

ePOap . . q-&Ll,

@...&Ll*

(27)

We note that just from (20) Baker was already able to derive bounds for the transcendence measure of e” and op with the usual conditions on (Y, /3 E A: leT - Cl > fpI”I”H,

laD-coJ

>H,

-co

In In Ho 7

where C,<s E A, H = H(C), Ho = H(
1.5. Methods of Proving Bounds for A, A,, and Ai. The inequalities (6)(8) and (19)-(26) are proved along the lines described in $1.3, except that instead of the maximum principle one uses Hermite’s formula (29) of Chapter 3. The sharpness of the resulting bounds depends on a good choice of auxiliary function f(z) and also on how far one can push the interpolation process. In every case the proof concludes with a contradiction: one proves that all of the coefficients of the auxiliary function are zero, although the function was chosen with nontrivial coefficients. Different techniques are used to achieve this. Sometimes one is able to obtain a system of linear equations in the coefficients with nonzero determinant. Several papers use induction on the number of logarithms, in which case the following lemma of Kummer plays an important role. Lemma w,...,QIm

4.2. Suppose that K is an algebraic number field of finite degree, E K, p is a prime number, and II& = K(oJ:‘~, . . . , a?-,). Then

either l& = l&((rzp) is an extension of II& of degreep, or else there exist y~Kandy,... , ~m-1 E %, 0 5 Vj < p, such that vm-1 a m = q “1 . . .(11,-l yp . Some deep theorems on the number of zeros of multivariable polynomials are used in the papers by Philippon and Waldschmidt and by Wiistholz where

188

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

Theorems 4.15 and 4.16 are proved. Such theorems are also an essential tool in various papers in which these results are strengthened and refined. 1.6. A Special Form for the Inequality. For certain applications it is useful to write inequalities for linear forms in logarithms or for the numbers (27) in a different form. In [1968b] Baker proved Theorem 4.17. Suppose that K. > m + 2 if all of the algebraic numbers are real, and K > m + 3 otherwise. Let bl, . . . , b, E Z, H = max lbjl, j = 1,. . . ,m, 6 > 0. If P,~l,...,wn

0 < (Yp *. . a$ - ,f3I < eedH ,

(28)

then H < max (c, In” H(P)) , (2% where c > 0 is an effective constant that depends on m, IS, n = max(deg(rr, . . . . . . ) dega,,degP), a = m=4lPI, IF’I), H(m), . . . ,H(sd, and 6. From Lemma 4.1 it is clear that under the conditions in Theorem 4.17 one can replace (28) by the inequality 0 < ]brlnar

+...+b,lna,

-lnp]

<e-‘OH,

where we agree to choose the argument of ,0 compatibly with Im(bi In al + ...+ b,lna,). Subsequent papers by Beukers, Sprindzhuk, Stark, and others lowered the value of IC, until finally in 1971 it reached K.= 1. Theorem 4.18. Suppose that In al, . . . , In cr, are fixed values of the logarithms o~(;Y~,...,(Y, EA. Letj3EA,n=degQ(ar ,..., crm,/3), and6>0. If the inequality O<]brlnar+~~~+b,lno,-lr~fi]<e-~~ holds, where bl, . . . , b, E Z and H = mal<j<,,, H < clnL@)

lbj 1, then it follows that

,

(30)

where c = c(m, n, 6, (~1,. . . , am, In (~1,. . . , In (Y,) is an efective constant. A similar theorem holds for cqh “‘CYy,b, -P. 1.7. Non-archimedean Metrics. In the work of many mathematicians, such as Brumer, Coates, Kaufman, van der Poorten, Sprindzhuk, and Yu, the theorems giving bounds for A, As, .4r, exp A - 1, exp As - 1, and exp A, - 1 have been generalized to non-archimedean metrics. For details, see [Yu 1989, 19901. Just as in the case of inequalities for archimedean metrics, these bounds have been applied to solve many problems in number theory. We shall discuss this in $2.

$2. Applications

$2. Applications

of Bounds

on Linear

Forms

189

of Bounds on Linear Forms

2.1. Preliminary Remarks. When studying certain problems one sometimes encounters upper bounds in various metrics for expressions of the form 6 = aa;’ . . . az

- @,r . . . bp ,

(31)

where cr, /3, ai, bj, xU, yv belong to specified sets. An example of this was the inequality (37) of Chapter 3. If the numbers a,/?, ai, bj,s,,yv have special properties - for example, if they are algebraic, or if they lie in U& - then 6 can be bounded from below using the theorems of $1. By comparing the upper and lower bounds for 6, we often obtain conditions on the terms on the right in (31). During the last 25 years many theorems have appeared whose proofs are based on the above line of argument. Here we will give only a few of these results. All of the results that follow are obtained in one way or another from bounds on linear forms in the logarithms of algebraic numbers. In some casesone can prove theorems that generalize or strengthen the theorems that were obtained in $5.6 of Chapter 3 using estimates of linear forms with two logarithms. 2.2. Effectivization of Thue’s Theorem. In 55.6 of Chapter 3 we already spoke about Gel’fond’s program. Now we are ready to show how this program is carried out. We begin by showing how the problem of estimating solutions of Thue’s equation reduces to Theorem 4.17. Let (X, Y) E Z2 be a solution to the equation f(X,Y)

= M,

M E z,

fc? Y) E axe, Yl,

M#O,

(32)

where f(z, y) is a form of degree n. We shall assumethat the leading coefficient is 1, i.e., f(X,Y)

= fi

(x - Jk)Y)

,

k=l

since the general case is easily seento reduce to this one. We shall also suppose that the form f(x, y) is irreducible, and that n 2 3. In particular, w(l), . . . , ~(~1 lie in ZA and have degree n. We set A(k) = x - J”)y

.

(33)

Suppose that ~1 of the numbers ~(~1 are real, and 2rz of them are complex, ri + 2~2 = n. By the Dirichlet unit theorem, the field lK = Q(w(l)) has r = ~1 + rs - 1 “fundamental units” Sii), . . . , 6(l) r such that any unit a(‘) of this field can be uniquely written in the form e(l)

= p(l)

(tp)zl..

. (p)zr

)

X1,.

. . ,z,

E z

)

190

Chapter 4. Generalization

of Hilbert’s

Seventh Problem

where p(r) is a root of unity. -If we let y (A denote the conjugate of y = y(l) E K = K(l) in the field K(j) = Q(&)), then @I,. . . , @ form a set of fundamental units of the field K (j). If T > 1, then there are infinitely many choices of a set of fundamental units of R. Let us number the conjugates u(j) in such a way that the first r-1 are real, and the (r-1 + j)-th and (~1 + ~2 + j)-th are complex conjugates, j = 1,. . . , ~2. Let et = 1 if ~(~1 is real, and et = 2 otherwise. Then the determinant

is nonzero, units, and only on n In what

its absolute value does not depend on the choice of fundamental R = [Al 2 ~0 > 0, where cc is an effective constant that depends and H(f(z, y)). R is called the regulator of K. follows we shall work with a system of T units of lK

#) ,...,rp

(34)

. . . (#‘)‘* = that are multiplicatively independent, i.e., the relation ( 7 qzl lforsr,.. . , z, E Z implies that zr = . . . = z,. = 0. If T > 0, there are infinitely many such sets; for example, any set of fundamental units is multiplicatively independent. The choice of units (34) will affect the constants in the theorems below. It is desirable for the heights of the units in (34) to be as small as possible. This is related to the ratio of the minors in

D = letlnl$)IIj,t=l,...(p to D itself. In this connection,

in [1969] Siegel proved

Lemma 4.3. There exists a set of multiplicatively “‘7 q. in K such that c

independent

units 771,. . .

j=l

where c is an effective constant

depending only on deg K, and R is the regulator

of K. Recall that for (Y E A the notation m means max loti) 1, where cr(“) runs through the conjugates of cr. In what follows ci will denote positive effective constants that depend only on the polynomial f(z, y). In particular, the c in Lemma 4.3 that bounds the units (34) is such a constant. It is fairly easy to prove that there exist br , . . . , b, E Z such that

c&%fll’n 5 where

IP@)l

I

C21w’n

,

(35)

52. Applications /A@)= A@) (QI,y

of Bounds on Linear Forms

. . . (qqv

for

191

t = l,.. . ,T .

Let H = lyF& PA * -One shows that there exist indices a and b such that IA(

5 c#pPH

>

pq

(36)

> C&t411hPH

.

(37)

Since n 2 3, there also exists a number Xtc) with c # a, c # b. If we eliminate X and Y from the system of equations (see (33)) x - (#)y

- x(4 = 0

x - ,J*)y

_ jj(b) = 0

x - &)y

- jj(c) = 0 )

we obtain: (

,W

_ ,,,W) x(c) + (,JV - w’4) ~(4 + (,(4

- J4)

x(b) = 0 .

Dividing through by X@) (~(~1 - w(“)) ~(~)/p(*), we find that

Let

&) _ JQ) ,Jb) P= &I - w(4*a ’

(38)

and lncri,... , In or, In /3 be the principal values of the logarithms. Then with a suitable choice of be E Z we obtain the following inequality from (35)-(38): Ibcln(-l)+bilnai+~~~+b,lna,-In/?1



-PI


-c.glH

.

We can now use Theorem 4.17 to arrive at the inequality H < ci0ln”L(/3)

.

A stronger inequality follows from Theorem 4.18: H < ~11lnL(/3) .

(39)

As a result of (38) and (35) one can find an upper bound for L(p). To do this we make use of Lemma 4.4. If p, 9 E A, deg 0 = nl, degp = n2, and deg Q(p, 19)= n, then qe f p) 5 3”IJ(e)“‘“‘qp)“~“”

;

192

Chapter 4. Generalization L(ep*l)

of Hilbert’s

5 2nqe)nlnlqp)n~n~

Since /.A@) and ~(~1 are algebraic

integers,

L(db'>,L(dC))

and since L(&))

5 ~13, it follows -w

From this inequality,

fort=l,...,

I

.

(35) gives the bound C12pq

;

from (38) and Lemma 4.4 that I

C141Jw16

.

(35), (36), and (39) we obtain the bounds H

pq

Seventh Problem

I

5~16

+C17lnIMI

,

l/~e+3+~17~~~~~)

C2pq

<

C18~~IClQ

n. It remains only to solve the system

after which

x - &)y

= x(l)

x - (J2)y

= x(2) 7

of equations

we finally obtain: IXI,lYl

I

C20 (IA(l)I

+ Ix(2)I)

It is easy to see that the same inequality the coefficient of xn in f(z, y) is not 1. 2.3. Effective deduces Theorem C22 = C22(cI)

Strengthening

5 C211MIC1Q

(with

of Liouville’s

.

different constants)

Theorem.

(40)

holds if

From (40) one easily

4.19. Let (Y E A, degcr = n 2 3. There exist effective constants > 0 and C23 = C23((Y) > 0 SUCh that

I I a - ;

> C22qC23--n

for any p E Z, q E N. Proof.

Let a,.?

+ . . . + a0 = a, fi

(z - &)

j=l

be the minimal

polynomial

of o = o(l), and set

f (x, y) = anxn + an-~xn--ly

+ . . * + aoyn = a, fJ (x - Gy) j=l

fhq)=M. Since f (z, y) is irreducible,

it follows

from (40) that

)

32. Applications

of Bounds on Linear Forms

193

We may suppose that ]CX- p/q1 < 1; then

Now

from which

the theorem

follows

immediately.

The inequality (40) and Theorem 4.19 were obtained in 1971 as corollaries of Theorem 4.18. However, the first (less precise) inequalities were found by Baker [1968b], who derived them from (29). The later refinements of (29) led to improvements on Baker’s original bounds for solutions of Thue’s equation and for ICY- p/ql. As noted before, the above constants cis, czi, ~22, and ~23 are effective. Various investigators - Baker, Gyijry, Kotov, Papp, Sprindzhuk, Stark, and others - have found explicit formulas for them. For details, we refer the reader to [GyGry 19801, [Sprindzhuk 19821, [Shorey and Tijdeman 19861, [Schmidt 19911, and [de Weger 19891. Here we shall only give two inequalities from [Baker and Stewart 19881. Let a E N, a # b3 for b E N. Let C be the smallest unit of Q(G) that is greater than 1. Set Cl =

C(501nw

7

c2 = 1012 In C .

Then 1)

ForanypEZandqENonehas

I I &G‘-

2)

g > cq+,

where

c = (34~1,

K. = 3 - ;

.

Ifz,yEZ,mEN,and X3

- ay3 = m ,

then max(]z],

Iv]) < cymcz

.

Unlike the similar inequalities in $33.5-3.6 of Chapter 1, which were proved using Pad& approximations, the bounds of Baker and Stewart were proved using estimates for linear forms in the logarithms of algebraic numbers. 2.4. The Thue-Mahler

Equation.

We already spoke of the equation

194

Chapter 4. Generalization

of Hilbert’s

Seventh Problem

in $10 of Chapter 1. The p-adic versions of effective bounds for the linear form A make it possible to obtain effective versions of Theorem 1.49. The first such results were due to Coates [1969, 1970a, 1970b]. Theorem 4.20. Suppose that f(x, y) E Z[x, y] is an irreducible form of degree n 2 3, N E Z, N # 0, P = {PI,. . . ,pS} is a fixed finite set of primes, and N = mM, m, M E Z, where all of the prime divisors of m are in P and CM, PI -..ps) = 1. l-f N,x, Y E Z, g.c.d.(x, y,n . . .pS) = 1, satisfy the equation f(x,y)

= N 9

then for any p > 0 one has max(]x],

]y]) < ce(‘“IMi)“, c = exp

v = %(s P Theorem 4.21. If x,y f (x, y) is greater than

(

(

n=n(s+l)+l+p,

2vaP2svn2 (1+

+ l)K’,

H(f F))

>

P = max(pr , . . . ,p,, 2) .

E Z, (x, y) = 1, then the greatest prime

1nlnX lO*n*(lnH(f)

divisor

of

l/4

+ 1) >

’

X = m@4 IYI) .

(41)

Coates obtained similar bounds for the equation y2 - x3 = k as a consequence of his p-adic version of Baker’s first estimate for A. The subsequent improvement of the bounds for A and their p-adic analogues led to refinements of Coates’ theorems. For details, see [Sprindzhuk 19821 and [Shorey and Tijdeman 19861. See also [Serre 19891. 2.5. Solutions in Special Sets. Let P be a fixed finite set of primes, and let S be the set of all rational integers whose prime divisors all belong to P (we also suppose that fl E S). We have already studied solutions of equations in S, for example, in the case of the equation x-y = c (see s4.2 of Chapter 1 and 55.6 of Chapter 2). There has been much work on solutions of equations or inequalities in rational numbers whose numerators and denominators (or only the denominators) have prime divisors that all belong to P. We encountered one such situation in 556.3 and 10.1 of Chapter 1. The notion of the set S can be extended to algebraic number fields, where instead of prime numbers one takes either irreducible elements of the field or else prime ideals. Many theorems have been proved about the solutions in S of various problems. One can find a great deal of information on this in the books [Schmidt 19911 and [Shorey, Tijdeman 19861.

$2. Applications

of Bounds on Linear Forms

195

In some cases one can approach these problems using bounds for linear forms in the logarithms of algebraic numbers. We illustrate this using theorems from [Tijdeman 1973b] and [Shorey, Tijdeman 19861. Theorem 4.22. Let P be a finite set of primes, and let n1 < n2 < . . . be the sequence of natural numbers all of whose prime divisors belong to P. Then

nj+l

- nj 2 nj(lIITIj)-c,

nj 2 3,

where c = c(P) > 0 is an effective constant. Theorem 4.23. Let K be an algebraic number field of finite degree over Q, let P be a fixed finite set of prime ideals of IK, and let S be the set of elements of & whose prime divisors all lie in P. Suppose that CY~,. . . , Q~ are distinct elements of IK, m,r,rl ,..., rnEN,m>3, bEI& and f(x, z) = (x - a12)”

* * * (x - cYn,Zp .

If

at least two of the rj are equal to 1, and if x,y equation fb, z) = bym 9

E &,

t E S satisfy

the

where msy(min(ord,x,ord,,y))

5 7,

then there exist a unit n E K and an eflective constant such that the heights of qx and qz satisfy the bound H(v)

+ H(v)

c = c(K, P,

f, b, m,

T)

5 c*

2.6. Catalan’s Equation. In [1844] Catalan conjectured that, except for 8 and 9, there are no cases of consecutive integers that are perfect powers of natural numbers. In other words, the equation xm-yn=l, in the four variables

X,Y,m,n

> 1,

(42)

X, Y, m, n E N has only one solution X=3,

Y=2,

m=2,

n=3.

(43)

Before 1976 only partial results had been obtained. In [1850] Lebesgue proved that there are no solutions with n = 2. In [1921] Nagell showed that (43) is the only solution with n = 3, and there are no solutions with m = 3; and in [1965] Ko Chao showed that (43) is the only solution with m = 2. A confirmation of Catalan’s conjecture came within reach after Tijdeman [1976b] proved that the solutions of (42) are bounded, i.e., max(X,

Y, m, n) < C ,

(44

196

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

where C is an effective constant. He obtained this result using bounds for linear forms in two or three logarithms. To prove Catalan’s conjecture it then remained to compute C and check the values of X, Y, m, n E N satisfying (44). The first numerical result was due to Langevin [1975/1976], who showed that X”

< exp exp exp exp 730

and

P(mn) < exp241 ,

where P(mn) denotes the largest prime divisor of mn. Without loss of generality one may assume that the exponents m = p and n = q are prime. In [1992] Mignotte proved that q < 1.31 x lol* .

p < 1.06 x 1026,

Various necessary conditions on p and q have been proved. The following criterion is due to Inkery [1964]. If p E 3 (mod 4), then either p*-’ E 1 (mod q”), or else the class number of Q(G) is not divisible by q. If p E 1 (mod 4), then either p*‘-l s 1 (mod q2), or else the class number of Q(e2”i/P) is not divisible by q. These conditions enable one to substantially reduce the work required to run through the 4-tuples that could be solutions to (42). However, it is not yet computationally possible to complete the proof of Catalan’s conjecture. More information about Catalan’s equation and its generalizations can be found in [Shorey and Tijdeman 19861. 2.7. Some Results Connected with Fermat’s Last Theorem. * We now describe some results that are related to Fermat’s conjecture that the equation

xn + yn = zn

(45)

has no solutions x, y, z E N for n E N, n 2 3. Proofs of the theorems below are given in [Shorey and Tijdeman 19861. Theorem 4.24. Let x,y, z be a solution of (45), where n > 2. If n is odd and z - y > 1, then z - y > 2n/n. Moreover, there exist effective constants cl and c2 such that l-cl(lnn)3/n.

IY-xl>z

7

I(z

-

y)

-

(y

-

x)I

>

zl-4w3/~

.

Theorem 4.25. Let B E N, and let S be the set oft E N all of whose prime divisors are 5 B. If the triple x,y,z is a solution to (45), and if at least one of the six numbers x,y,z,y - x, y + z,x + z (respectively, at least one of the first four of these numbers) lies in S when n is odd (resp. even), then n + x + y + z < cs, where cs = Q(B) is an effective constant.

Suppose that F(X,Y) is a quadratic form over Z, n > 2, and the triple x, y, z is a solution to (45). If at least one of the numbers F(x, y), Theorem

4.26.

* The results in this section have been superseded by [Wiles 19951.

32. Applications

of Bounds on Linear Forms

197

F(x,z), F(y,z) belongs to the set S in Theorem 4.25, then x + y + z < ~4, where c4 = cq(n, B, F) is an eflective constant. 2.8. Some Other Theorem Q(m,...

Diophantine

Equations.

We shall give just a few examples.

4.27. Let m 2 3, crl, . . . , (Yrn,pE , am, p) . Then the equation

z.4, ai # aj for i #j, K =

(X - a1Y) * * * (X - amY) = ji has only finitely many solutions x,y E Zn<, and H(x) by an efective constant c = C(CQ,. . . , CK~,II).

(46)

and H(y)

are bounded

A non-effective version of this theorem was proved by Siegel (see 55 of Chapter 1). Note that equation (46) is a generalization of Thue’s equation (3% Theorem 4.28. Suppose that the polynomiaI P(z) E Z[Z], degP = n, has at least two simple roots, and let a E Z, m E N, m 1 3. Then there exists an efective constant c = c(m, n, H(P)) such that any solution x, y E Z of the Diophantine equation P(X) = aY”

satisfies the bound 1x1+ IYI I c * These two theorems of Baker are proved in his book [1975a]. A non-effective version of Theorem 4.28 was proved by Thue [1917], who obtained it as a corollary of Theorem 1.11. The proof of Theorem 4.28 implies, in particular, that any solution of the famous Diophantine equation x2 - Y3 = k,

kc&

k#O,

(47)

satisfies the bound 1x1+ IyI < exp (10106]k]104) . This inequality of Baker was substantially improved by Stark [1973b]; and Sprindzhuk [1982, p. 1491later obtained the bound

I4 + IYI < exp (4 In60 + 14)) . A special caseof (47) was mentioned as early as 1621 by Bachet. The history of this equation is discussed in [Dickson 1919, Chapter 201and [Mordell 1969, Chapter 261. We now give two theorems from [GyBry 19801and [Gyijry and Papp 19831. They provide an effectivization of Schmidt’s Theorem 1.46 in certain cases. Theorem 4.29. Let (Y2,..., (.Y~EA,IK=Q(cQ ,..., at),degQ(cri)=ni>3, degK = n = 122-aant, a E Z, a # 0, Ial, lY!EJ 5 A, i = 2,..., t, M E Z. Then any solution (xl,. . . , xt) E Zt of the equation

198

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

a Norm(zl + ~2~x2+ . +f + QCQ) = M

(48)

satisfies the bound lyzt -where R

is

1~1 < exp(cR(R + ln(AIMI))

ln2(R + e)) ,

the regulator of IK and c = c(n) is an effective

constant.

Theorem 4.30. Let (~2,. . . , at E A, a E Z, a # 0, (al, lTiSJ 5 A, I& = Q, l& = JK+l(ai), [l& : l&-l] 2 3, i = 2,. . . , t. Then any solution (xl,. . . , xi) E Zt of (48) satisfies the bound

l’=zt -- I4 < < exp QIq3n2-9n+6

1q$n~-$n+3

(

+ ln(AIMI))

(

(ln ID~)3nS-gnz+6’+1) ,

where D is the discriminant of the field Q(a2,. . . , at) and co = co(n) is an effective constant. In [Gybry 19801 similar results are obtained for some special cases of the generalized Thue-Mahler equation Norm(zr + zpop + . . . + ztat)

= m$

. . .pf’ ,

where m E Z, {PI,... ,p8} is a fixed set of primes, and we are interested in solutions (21,. . . , zt, 21,. . . , 2,) E ZFs. Here is another example from [Shorey and Stewart 19831. Theorem t

4.31.

Let a, b,c,d E Z, acd(b2 - 4ac) # 0. Ifx,y,

t E Z, 1x1> 1,

> 1, and ax2t + bxty + cy2 = d ,

then

I4 + IYI + t I c , where c = c(a, b, c, d) is an effective constant. We cannot drop the assumption that t > 1 in Theorem 4.31, since when = 1 there may be values of d for which the equation has infinitely many solutions (see the discussion of Pell’s equation in $2.2 of Chapter 1). In $5.1 of Chapter 1 we gave Siegel’s theorem on integer points on algebraic curves of genus 1. This was a non-effective theorem. In [1970] Baker and Coates proved an effective version of this theorem. t

Theorem 4.32. Suppose that F(X, Y) E Z[X, Y] is irreducible of degree n and height H, and suppose that the curve F(X, Y) = 0 is of genus 1. Then any solution x, y E Z of the Diophantine equation

F(X,Y) satisfies the bound

= 0

$2. Applications max(]z],

of Bounds on Linear Forms

]y]) 5 expexpexp

(

(2H)‘““‘o)

199

.

2.9. The abc-Conjecture. For a, b, c E N let G(a, b, c) denote the squarefree part of abc, i.e., G=G(a,b,c) = n p. plabc,p a prime Masser and Oesterle conjectured: If a + b = c with (a, b, c) = 1, then for any E > 0 there exists 70 = TO(E) such that c < yoG’+& . (49) A proof of this conjecture would bring us closer to solving several problems in number theory. For example, let a = xn, b = yn, c = P, n, x, y, .z E N, n 2 3. If abc-conjecture is true, then any solution x, y, I of the Fermat equation xn+yn=zn would satisfy the bound c = Z” 5 -yo(~)G1+E . Let E = 0.25. Since G= n

p 5 xyz < z3 ,

PbW

it follows that zn

5

-y~(1/4)z3+"~75

)

and so Z

n-3.75 < 7)(1/4) .

This would mean that Fermat’s Last Theorem holds for n > no, and for every n E N, 4 < n 5 no, one would obtain an upper bound on the value of z in a solution. This bound would be effective if 7s is effective. It should be noted that for the purposes of Fermat’s Last Theorem it suffices to have c < ylG’rather than (49). In [1991] Stewart and Yu Kun-rui proved Theorem 4.33. There exists an effective constant 7 > 0 such that c < exp G2j3 +y/lnlnG (

>

.

2.10. The Class Number of Imaginary Quadratic Fields. In 55.6 of Chapter 3 we mentioned that bounds for linear forms in three logarithms of algebraic

200

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

numbers can be used to prove that there is no tenth imaginary quadratic field of class number one. In [1969] Bundschuh and Hock implemented the idea of Gel’fond and Linnik [1948], and used Baker’s inequality (6) to prove that Die does not exist. Now suppose that the field Q(a), w here d E N is not a perfect square, has class number 2. It is known that in that case 0 < ]&lnai

+/3zlnaz

+Pslnas]

< e-O.l&,

where cri and ~2 are the fundamental units of certain real quadratic fields, CQ= -1, pi, P2 E Z, and /3s = 64a/31. Here Ipi] and I/32] are bounded by d”, where c is an effective constant. After strengthening Theorem 4.17, Baker and Stark were able to obtain an effective upper bound for d. Their calculations showed that d = 427 is the largest discriminant of a field of class number 2. For details, see [Baker 1971a, 1971b, 1975a], [Baker and Stark 19711, [Stark 1971, 19721. The paper [Baker and Schinzel 19711is also related to these questions. Using completely different techniques (from the theory of elliptic curves and L-series), Goldfeld [1976] and Gross and Zagier [1986] proved an effective bound for the discriminants of imaginary quadratic fields of any given class number. 2.11. Applications in Algebraic Number Theory. Let (Y E A, degcr = n > 1, and let o(l) = o,. . . , ocn) be the conjugates of (Y. The disctiminant of (Y is the number D(a) = n (Q(i) - (y(j))2 . l
Obviously, D(a) E Q; and if (Y E ZA, then D(cY) E Z. If p = cr + r for T E 0, then p and cr have the same discriminant. Let B(a)={/3: B~(a)={p:

p=(Y+r, p=a+m,

rEQ}; 7nEZ).

Theorem 4.34. Let D E Z, D # 0. There are only finitely many classes &(a) of algebraic integers whose elements satisfy D(a) = D. Every such class l&((r) contains an element y for which

H(7) I c ,

(50)

where c = c(D) is an effective constant. We fix D E Z. Let M = M(D) be the set of algebraic integers y for which (50) holds with c = c(D) and also D(y) = D. It is known that for any cx E ZA one has dega I &(l + In ID(c .

$2. Applications of Bounds on Linear Forms

This implies that the degree and height of any element of M(D) hence, M(D) contains only finitely many numbers.

201

are bounded;

Theorem 4.35. Let D, M E N. There are only finitely many algebraic integers 8 for which INorm@] 5 M . ID( L D, Any such 6’ satisfies the bound

where Q = %(D, M) is an effective constant. Theorems 4.34 and 4.35 are proved in [Gyory 19801. The next theorem is due to Anfert’eva and Chudakov [1968]. Theorem 4.36. Let D be the discriminant of an imaginary quadratic field lK of class number h, let Al,. . . , Ah be the ideal classes of I& let M, be the smallest norm of an ideal in A,,, and set M = max M,. There exist effective constants cl = cl (h) > 0 and c = c(h) > 0 such that

m M 2 clIn D(lnln

0)”

*

2.12. Recursive Sequences. By a recursive sequence we mean a sequence {un} that satisfies an equation of the form Un+k

n=O,l,...

= uk-lZLn+k-l+~~~+~lUn+l+~OZln,

.

(51)

Such a sequence is uniquely determined by its initial conditions, i.e., by the Ic-tuple ~0, . . . ,u~-~. Suppose that a~,. . . , ok-1 are complex numbers, and define the characteristic polynomial of the sequence to be P(Z) = Zk - ~k-lzk--l

- “* - a0

= fi(Z-Oj)Uj,

ei # eh

for i # h.

j=l

Then the general solution of (51) has the form un = 2

Rj(n)ej”

,

(52)

j=l

where the Rj (z) are arbitrary polynomials of degree Vj - 1 with complex coefficients. A particular solution is determined by initial conditions, which enable one to solve for the coefficients of the polynomials Rj(z). If as,. . . , a&i, uc, . . . , u&i are algebraic, then so are all of the u,. It is frequently possible to prove certain properties of the sequence {un} using bounds for the differences between terms in the sequence, i.e., bounds

202

Chapter

4. Generalization

of Hilbert’s

Seventh Problem

for the differences between exponential expressions as in (5) of Chapter 1. This sometimes reduces to estimating linear forms in logarithms. We have already examined similar questions in connection with bounds for linear forms in two logarithms (e.g., Theorem 3.24). We now give some results that can be proved using bounds for linear forms in several logarithms. Suppose that k = 2, i.e., un+2 = uun+i + bun, where a, b,ue,ui E Z, ~ue~+~ui~>0,a2+4b#0.Then U n-

-A#+Bpn,

A=

‘LLOP p-a

Ul

w ’

B=

p-cl!

uoa ’

where Q and /? are the roots of the polynomial .z2 - az - b. Suppose that I4 2 PI, AB # 0, and (Y/P is not a root of unity. Theorem 4.37. There exist effective constants c and co depending only on a, b, UO,UI such that

Iu, - 21,) 1 Iapqm

+ 2)-cl++2),

mfn,

M=max(m,n)>c0.

Let M E Z, M # 0. We let P(M) denote the largest prime divisor of M, and let Q(M) denote the squarefree part of M (i.e., the product of all primes dividing M). Let v = dega (thus, Y = 1 or 2). Theorem 4.38. There exist effective constants cl, ~2, cg, and ~4, depending on a, b, uo, ~1, such that

P(u,)

2 Cl (+Jl’++l)

)

n 2

~2 ,

Q(4

2

(~3%)

,

n2

c4

and exp

.

Now suppose that k 1 2, and the sequence {un} is given by algebraic initial conditions and the relation (51) with algebraic coefficients. In (52) suppose that Pll 2 18212 **- L l&l and p11 = . . . =

l&l

>

l&,lI

*

Theorem 4.39. Suppose that r = 3, 101I > 1, and at least one of the numbers e1/e2, e1/e3, e2/e 3 is not a root of unity. There exist efective constants cg, cg, ~7, and cfj, depending only on the coeficients in (51) and the initial conditions, such that lu,l 2 14 in exp(-c5 ln2 n), nlcs;

Iu, - urnI > ~el~~exp(-c71n2nln(m

+ 2)),

n>m,

n>c8.

$2. Applications

of Bounds

on Linear

Forms

203

Theorems 4.37 and 4.38 are due to Shorey, and Theorem 4.39 was proved by Mignotte, Tijdeman, and Shorey. Proofs are given in Chapters 3 and 4 of [Shorey and Tijdeman 19861, where one can also find discussions of work on this question by Beukers, GySry, van der Poorten, Stewart, and others. Stewart’s survey [1985] should also be mentioned. 2.13. Prime Divisors of Successive Natural Numbers. For n E N the numbers 2n - 1, 2n + 1, and 2n are pairwise relatively prime. Hence, if n > 1, each of these numbers has its own prime divisors. For n E N let g = g(n) denote the greatest natural number such that there exist distinct primes pi, . . . ,p, with ptl(n + t), t = l,..., g. Several papers have been written giving lower bounds for g(n). Using estimates for linear forms in the logarithms of algebraic numbers, Ramachandra, Shorey, and Tijdeman [1975] proved that g>c

( > Inn lnlnn

3

n>3,

’

where c > 0 is an effective absolute constant. In (19761 the same authors proved that if k E N satisfies the inequality k 5 exp(ce&%)

,

where ~0 is an effective absolute constant, then there are at least k distinct prime divisors of the number (n + 1) . . . (n + k). Several theorems have been proven about the prime divisors of the product (n + 1) . . . (n + k) and the prime divisors of a product of successiveterms in a general arithmetic progression, i.e., (a + b)(a + 2b) . . . (a + kb), a, b E N. Once again we refer the reader to [Shorey and Tijdeman 19861for more information about this. 2.14. Dirichlet Series. We now give one of the results proved in [Baker, Birch, and Wirsing 19731. Suppose that q E N, (q,cp(q)) = 1, x(n) is a character modulo q, and

Wx)=C,,--

O” x(4

n=l

If none of the characters x is the principal character, then the numbers L( 1, x) are linearly independent over A. This paper also proves more general results about the series

f(n)7 L(s) =cO”y$n=l

.-

where f(n) is a periodic function that takes algebraic values on Z. In particular, it is pointed out that in certain casesone can use Theorem 4.9 to obtain the inequality

204

Chapter 4. Generalization

where H is the maximum period of f(n)).

of Hilbert’s

Seventh Problem

height of the numbers

$3. Elliptic

f(l),

. . . , f(q)

(here q is the

Functions

3.1. The Theorems of Baker and Coates. Just as the solution of Hilbert’s seventh problem stimulated work on the properties of elliptic functions, the work of Baker similarly led to new theorems in this area. We shall use the notation introduced in 53 of Chapter 3. In this section we shall assume that the @functions have algebraic invariants. Theorem 4.40. Let q and v’ be the qua&periods corresponding to the periods w and w’ of the Weierstrass g-functions ~(2) and p1 (z), and let (Y, /3, y, 6 E A. Then the number X=aw+pw’+yq+6q’ is either zero or transcendental. Theorem 4.41. Let CEO,(~1, (~2 E A, QO # 0, H = max H((Yo, CQ,CXZ), n = deg Q(~o, al, 4. Then Ia0 + cylwl + where K is an absolute constant. Theorem

constant

a2w2I

> ce-'"x

H ,

and c = c(gg, gs, WI, ws) > 0 e’s an effective

4.42. Let a E A, (Y # 0. Then Ip(a)I

where K. is an absolute constant.

constant

< Gel”= H(a) , and ~0 = Q(g2, gs, ~1, ~2) is an effective

Proofs of these theorems of Baker and related results can be found in [Baker 1969c, 1970a, 1970b, 1975a]. Many of these theorems were later refined and generalized. For example, in [1971a] Coates proved that one can add the term ET, E E A, to X in Theorem 4.40. .3.2. Masser’s Theorems. We say that (Y E Cc is an algebraic point of the Weierstrass pfunction if p(o) E A or if (Y belongs to the period lattice R of p(z). If Q(Z) h as algebraic invariants (this is the case we are considering in this section), then it follows from the addition theorem ((8) of Chapter 3) that for r E Q and p $ Q either both &?) and p(rp) are algebraic, or else both are transcendental. If w E 0 and w/2 $ R, then @‘(w/2) = 0, and so it follows from (7) of Chapter 3 that @(w/2) E A.

$3. Elliptic

205

Functions

The first bounds for linear forms in an arbitrary number of algebraic points appeared in [Masser 19751.Let r be the ratio of the two fundamental periods of P(Z), and let K = Q(7) if r E A and K = Q if 7 is transcendental. Theorem 4.43. Let ul, . . . , urn be algebraic points of p(z) that are linearly independent over K, and let c > 0. Then ~1,. . . , u, are linearly independent over A, and the following inequality holds for any pi,. . . , Pm E A, max ]oj ] > 0, degpj 5 n, H(Pj) 2 H, j = 1,. . . , m: l/3lul + . . . + ,¨ > cemHE, wherec=c(g2,g3,u1,...

(53)

, u,, c, m) > 0 is an effective constant.

Masser also proved that if one drops the condition that ~1,. . . , u, be linearly independent over K, then the linear form in (53) is either zero or transcendental (see Appendix 3 of [Masser 19751). In [1977b] Masser proved the following relations between the A-dimensions of certain vector spaces: dim{w,uz,w,m,~)

= 2dim{w,a}

+ 1

(the right side is clearly equal to 3 if Q(Z) h as complex multiplication, otherwise), dim{Lw,w,w,r/2,~)

= dim{w,ww,m,~}

dim{w,m,w,m) dim{l,tii,Wz,qi,n2}

and 5

+ 1,

= 2dim{w,a), = 2dim{wi,wa}

+ 1.

3.3. Further Results. The next theorem, which strengthens Theorem 4.43, is due to Anderson [1977]. Theorem 4.44. Suppose that p(z) has complex multiplication field K, ul,...,u, are algebraic points of g(z) that are linearly dent over !K, n E N, and K. > m + 1. There exists an effective u,) > 0 such that the following inequality c = c(~,n,g2,g3,u1,--., & E A, degpj 5 n, H(/?j) 5 H, H 2 3, j = any Po,Pl,..., IIXiXl~jl > 0: po

+

PlUl

+

*. . +

Pm u m I>ce

- In H(ln

In H)=

over the indepenconstant holds for 1,. . . , m,

(54)

In [1990b] Hirata-Kono improves and generalizes this result. She proves that, under certain conditions, if uj is an algebraic point of pj(z), then (54) holds with K.= m + 1. She also shows how the bound depends on the heights of pj(uj) and the degree of the field generated by the /3j and pj(Uj). Note that Theorem 4.44 implies, in particular, that the numbers 1 and uj are linearly independent over A. In the complex multiplication case this qualitative result was proved earlier by Masser [1976]. In [Bertrand and Masser

206

Chapter 4. Generalization

of Hilbert’s

Seventh Problem

198Oa] a different method (see [Schneider 19411) is used to prove an analogous theorem in the non-complex multiplication case. These results are elliptic analogues of Baker’s Theorems 4.4 and 4.7. 4.45. Suppose that p(z) does not have complex multiplication. Let be algebraic points of p(z) that are linearly independent over Q.. Then 1,2~~,...,2d~ are linearly independent over A. Theorem

w,**.,%z

Nagaev’s paper [1977] is also concerned with these questions. 3.4. Wiistholz’s Theorems. The results given below appeared in [Wiistholz 1984a]. Let c(z) be the Weierstrass C-function corresponding to P(Z), let w be a nonzero period, and let n = q(w) = C(z + w) - C(z) be the corresponding quasiperiod. Let 2~1,. . . , urn be algebraic points of p(z) that are not in the period lattice, and set X(uj, w) = wC(uj) - VZL~.Note that the function UC(Z) nz is periodic with period w. Theorem

4.46.

If

x = crlX(U1)w) + **. + wn~(%, w>+ Pw + Porl# 0 , whereal,...

,(~~,/I,/30 E A, then X is transcendental.

An important consequence of this theorem is that the periods of elliptic integrals are transcendental. Theorem 4.47. Let R(x, y) E A(x, y), and let y be a closed contour on the Riemann surface of the curve y2 = 4x2 - g2x - gs, where g2,gs E A. Then

JYR(x, Y&

is either zero or a transcendental number. This result follows from Theorem 4.46 because the integral in Theorem 4.47 can be written as a linear combination with algebraic coefficients of numbers of the form X(ui, w), . . . , X(u,,,, w), w, 71,where uj are algebraic points of p(z). In [Laurent 198Oa] the last theorem was proved under the additional assumption that all of the residues of the differential form in the integral are rational. Laurent also proved some other special casesof the theorem. The next theorem answers the question: Under what conditions can the linear form X in Theorem 4.46 vanish? 4.46. Let T be the number of elements in a maximal subset of . . , urn} that is linearly independent over Q. Then there are exactly r + 3 numbers in the set { 1, Xi, . . . , A,,, , w, 7) that are linearly independent over Q. Theorem

{w,.

54. Generalizations

of the Theorems in $1 to Liouville Numbers

$4. Generalizations of the Theorems to Liouville Numbers

207

in $1

4.1. Walliser’s Theorems. The results in Theorems 4.4 and 4.6 remain valid if the exponents have “good” approximations by algebraic numbers. We have already encountered such generalizations in $6 of Chapter 3. The next two theorems were proved by Walliser [1973a]. Theorem 4.49. Let al,. . . , am be distinct algebraic numbers not equal to 0 or 1. Suppose that the numbers 1, @I,. . . , P,,, are linearly independent over Q, n E N, IS > m + 1, and for arbitrarily large H the inequality

has solutions the number


5 n, H(O)

5 H, j = 1,. . . , m. Then

cqA “‘Cl, &n

is transcendental. Theorem 4.50. Let (~1,. . . , (Y, be distinct algebraic numbers not equal to 0 or 1. Suppose that /30,/31,..., /3,,, E C, & # 0, n E N, n > m+ 1, and for arbitran’ly large H the inequality

has solutions the number


m. Then

. ..c@m m

is transcendental. 4.2. Wiistholz’s Theorems. Suppose that E > 0, n E N, and the logarithms ofal,..., a, are linearly independent over Q. Further suppose that there exist a strictly increasing sequence of natural numbers Aj for which the ratios In In Aj+l / In In Aj are bounded from above and an infinite sequence of vectors (orj, . . . , om,j) E A.- with H(ai,j) 5 Aj and degaij 5 n that satisfy the inequality Ial -

'Yl,jl

+

. . . +

la, -

CXm,j

1<

e-(lnAj)“2-“+e

,

Theorem 4.51. For any ~1,. . . ,x~ E Z with 0 < rnaxlzi] 5 X, X 2 2, and for any HO = HO(e,n,m,al,..., a,) one has

208

Chapter

Ial -GI

4. Generalization

+ .**+ 1% -&I

corollary

of Hilbert’s

Seventh Problem

+ 1211naI + . . . +Zmlna,I

> ~-(lnH)“‘-*+~

.

4.1. If

lal-~l;I+...+la,-~,I<X-

(In H)“2-“+c

,

then (21 In al + . . . + xm ln urnI > x-(‘“w2-n+c

.

This theorem was proved in [Wolfart and Wiistholz 19851. In [1979] Wiistholz obtained a lower bound for the following sum when al,. . . , a,, bl,. . . , b, are Liouville numbers: lal-all+...+lu,-a,l+Ibl-pll+... -.a+ lb, -/3,,J + ~u;~Yz~ whereal ,...,

a,,&

,...,

&,+ycA.

---/I,

51. E-Functions

209

Chapter 5 of Analytic Functions That Satisfy Differential Equations

Values

Linear

51. E-Functions 1.1. Siegel’s Results. In [1929/1930] Siegel studied the transcendence erties of the values at algebraic points of the function KA(Z)

= 1+ 2 (-lY z 272 n=l n!(X + 1). *. (A + n) 0 5 ’ x # -1, -2,.

and its derivative K:(z) for rational function Jx(z) as follows: Jxb) and it satisfies

prop-

=

the differential

.. )

A. This function

1 r(x+l)

z k(z) 05

is related to the Bessel

;

equation

2x + 1 y” + --yy’+y=o. Before Siegel’s work, Legendre, Hurwitz, Stridsberg, and Maier had studied the values of functions closely connected with KA(z) (see 54 of the Introduction to [Shidlovskii 1989a]). Siegel credited Maier’s work with stimulating him to develop a method that would make it possible to obtain much more general results. In particular, Siegel was able to prove the following theorems. Theorem 5.1. If X is a rational number not equal to half of an odd number, and if [ is a nonzero algebraic number, then the numbers Kx(c) and Ki(<) are algebraically independent over Q. Theorem 5.2. Suppose that none of the rational numbers Xl,. . . , A,, 1, are equal to half of an odd number. Further suppose that Xi f

Xj

4 Z,

l
n 2 1, are nonzero algebraic numbers a~dG,...,&, tinct. Then the 2mn numbers

Kxi (<j>, Kii (&I> are algebraically

independent

m 2

i=l,...,

m,

whose squares

j=l,...,

are dis-

n,

over Q.

Theorem 5.3. Let < be a nonzero algebraic number. For any s 2 1 and H 2 1 and for any polynomial P E Z[xl, x2], P $0, such that

210

Chapter

5. Functions

That

Satisfy Linear

deg P < s,

Differential

Equations

H(P) 5 H ,

one has IP(Jo(O, J;(t))1 L ~~~~~~~~~~ , where h is the degree oft

and LTis a constant depending only on [ and s.

Theorems 5.2 and 5.3 are generalizations of Theorem 5.1. From the relations K-r/2(~) = cosz, Kl_,,,(z) = - sinz, and Kx-1 = KA + &Ki 1 K’ X-l = -zT; = Kx ,

,

Kx = -aK’ .3 Ki = &KM 1

X-l

7

(1)

4x=K, + F

X-l

1

it follows that for any n E Z the functions Kn+1,2(z) and K;+,,,(z) are algebraically dependent over C(z), and their values at a nonzero point 5 E A are algebraically dependent over Q. From (1) it also follows that if Xr - X2 E Z, then the functions Kxl, Kil, Kx2, Ki2 are linearly dependent over C(z), and their values at a nonzero point E.E A are algebraically dependent over Q. One also has the relation (see 57 of Chapter 9 of [Shidlovskii 1989a]) KA(z)KI_,(z)

- K~(z)K-~(z)

- FKA(z)Kmx(z)

+ $

= 0,

Kx,(<), from which, using (l), we see that the numbers Kxl (<), Kil(<), K!J<) are algebraically dependent over Q whenever Xr + X2 E Z. Thus, the conditions in Theorems 5.1-5.2 are necessary as well as sufficient. Theorem 5.1 implies that, under the conditions in the theorem, the continued fraction

is a transcendental number. In particular, the number . Jo(2i) ,&-zz$J;(2i)

.Ko(2i) =1+!!+!!+Y+ KA(2i) 12 13

14

...

is transcendental. Several other results are stated without 1929/1930].

proof in [Siegel

1.2. Definition of EFunctions and Hypergeometric E-Functions. Siegel’s paper [1929/1930] was the point of departure for an entire branch of transcendental number theory. Recognizing that his ideas would apply in great generality, Siegel introduced a class of entire functions which, in his opinion, was the class to which one could apply the methods that he used to prove Theorems 5.1-5.3. He called these functions “Efunctions,” presumably because in some sensethey bear a resemblance to the function et = Cz=, 5.

fl.

Defhition

E-Functions

211

5.1. An analytic function f(z) = 2 Cn$ n=O

is said to be an E-finction

if:

1) all of the cn lie in an algebraic number field IK of finite degree; for any E > 0 one has

2)

pJ 3)

= O(P)

as

n + 0;) )

where m denotes the maximum modulus of the conjugates of a; for any E > 0 there exists a sequence of natural numbers q1, qz, . . ., with q,, = O(rP), such that for all n qncj

E ZK

for

Oljln.

The class of E-functions clearly contains all polynomials with algebraic coefficients, as well as the functions e”, sin Z, and cosZ. The Efunctions form a ring that is closed under differentiation, integration from 0 to Z, and the change of variables z C) oz, (Y E A. The generalized hypergeometric functions with rational parameters give many examples of Efunctions. Let aI ,..., al,br,. . . , b, be complex numbers not equal to 0, -1, -2, . . ., and let t = m - 1 > 0 (the case 1 = 0 is allowed). The generalized hypergeometric function

~+lFm

(

l,al,...,al >

h,

b2,.

. . ) briZ

=

2

(4

n=O

(bl),

* * * (4z * * * (b&

Zn ’

where the symbol (a), is defined by setting (ajo = 1,

(cx)~ = a(a + 1). . . (0 + n - 1)

for

n= 1,2,...

,

satisfies the differential equation

j&5 i=l

+ bi - 1) - z b(6 + ai)

y = (br - 1) . . . (b, - 1) ,

i=l

where 6 = z& (see [Luke 19691,formula (5) of Chapter V, $5.7.1). When 1 is added or subtracted to the parameters, the hypergeometric function behaves as follows (formulas (9)-(10) of $5.2.2 in Chapter V of [Luke 19691):

212

Chapter 5. Functions That Satisfy Linear Differential

Equations

One has similar identities when one of the ok is decreased by 1 or one of the bk is increased by 1. If t = m - 1 > 0, the function

is an entire function of order l/t. In transcendental number theory it is customary to make the change of variables z I+ (z/t)t, so that the resulting function has order 1 and type 1. In particular,

The function ‘+lFm satisfies

the differential fi(6

equation + tui)

+ t(bi - 1)) - z’ f&5

y = tm(bI - 1) +. . (b, - 1) .

i=l

i=l

In [1929/1930] Siegel proved that if ui, bj E Q, and if the bj are not equal to negative integers, then ‘+lFm

l,Ui,...,Q bI,b2,...,b,’ (

is an E-function; and, in fact, in requirements E-function one can replace O(nEn) by O(P) c that depends on the ui and bj. A function l,w

‘+lFrn

,...,a1

( bl,b2,...,b,’

z t (0t 2) and 3) of the definition of an for a sufficiently large constant

z

t

0t

with these conditions on the ui and bj is called a hypergeometric E-function. In [1949] Siegel conjectured that any E-function that satisfies a linear diflerential equation with coeficients in C(z) can be expressed as a polynomia! with algebraic coeficients in z and a finite number of hypergeometric E-functions

51. E-Functions

213

or functions obtained from them by a change of variables of the form z I-+ az for cr E A. This conjecture still has not been either proved or disproved, even in the case of an Efunction that satisfies a first order non-homogeneous linear differential equation. In [1981] Galochkin proved the following criterion for a hypergeometric function to be an Efunction: the function ‘+lFm with parameters

ai, bj E Cc satisfying

ai, bj # 0, -1, -2,. . . ,

the conditions l
ai # bjv

l<j<m,

is an E-function if and only if all of the ai and bj are algebraic and the irrational numbers among them (if there are any) can be divided into pairs . , (ai,, , bjU) for which (ai,,bj,),.. ai, - bj, E N,

s=l,...,u.

The necessity of this condition for the function i FI proven much earlier by Sprindzhuk [1968].

(

A:1 ; z

>

had been

1.3. Siegel’s General Theorem. In [1949] Siegel proved the following general theorem on algebraic independence of the values of Efunctions. Theorem 5.4. Suppose that the E-functions fl(z), . . . , fm(z) form a solution of the system of homogeneouslinear differential equations Y; = 2

Qki(Z)Yi,

k=1,2

,...,

m,

(3)

i=l

where Qk:i(z) E C(z). Further supposethat for any N E N the (“LN) products of powers f,“l(z>***fkm(z,,

O
kl+‘**+k,
(4

satisfy a certain normality condition (see 52 below). Then for any algebraic number Q not equal to zero or a pole of any of the Qki the numbers

are algebraically independent over Q. The Lindemann-Weierstrass theorem follows easily from Theorem 5.4. Siegel also proved Theorems 5.1 and 5.2 as corollaries of this theorem. However, the normality condition (see52) turned out to be so difficult to verify that during the next 40 years (until [Beukers, Brownawell, and Heckman 19881,see

214

Chapter 5. Functions That Satisfy Linear Differential

Equations

$3 of this chapter) not a single paper was published that applied Theorem 5.4 to a set of E-functions that includes the solution set of a third order linear differential equation. 1.4. Shidlovskii’s Ji’undamental Theorem. In [1954, 1959b] Shidlovskii was able to replace the normality condition in Theorem 5.4 by a simpler irreducibility condition (see $2). This enabled him to obtain many new results for hypergeometric E-functions that satisfy higher order differential equations. In [1955a, 1959a] Shidlovskii proved a definitive result (his First Fundamental Theorem) that reduced the general arithmetic problem of proving algebraic independence of the values of E-functions to the problem of proving that the functions themselves are algebraically independent. Theorem

5.5. Suppose that the E-functions fl(Z),...,fmb),

form a solution

of the system

of linear differential

y; = Q/co(z) + 2

(5)

m21,

Qki(z)~i,

k=l,2

equations ,...,

m,

(6)

i=l

where Qki(z) E C(z). Let a be any algebraic number not equal to 0 or a pole of one of the functions Qki(z). Then the numbers fl(~),...,fm(Q)

are algebraically independent over Q if and only if the functions braically independent over Q(z).

(7) (5) are alge-

It is a difficult problem - and in many cases an unsolved problem to prove that the solutions of a linear differential equation are algebraically independent. However, in some sense Theorem 5.5 takes care of the numbertheoretic part of proving algebraic independence of the values of E-functions at algebraic points. To illustrate the uses of Theorem 5.5, we mention the following result from [Shidlovskii 19611 that generalizes the Lindemann-Weierstrass theorem: Suppose that the transcendental function f(z) is a solution of the linear differential equation Y’ + P(.z)Y = q(z), p(z),q(z) E C(z) > and 51,. “, Em are nonzero algebraic numbers that are not equal to singular points of P(Z) 0~ q(z). If 6, . . . , trn are linearly independent over Q, then

are algebraically independent over Q. If, in addition, one knows that there is no rational function that solves the differential equation, then the assumption

52. The Siegel-Shidlovskii

Method

215

of linear independence over Q can be replaced by the condition that f$, . “, c m merely be distinct. To prove this theorem it suffices to show that the functions f(&~), . . . are algebraically independent over C(z) and then use Theorem . . * 7f(&r) 5.5 with z = 1.

§2. The Siegel-Shidlovskii

Method

2.1. A Technique for Proving Linear and Algebraic Independence. Siegel was the first to understand that one can prove that a set of numbers wr , . . . , w,,, is linearly independent not only by using simultaneous rational approximations, as Hermite had done, but also by constructing a complete system of linearly independent linear forms with integer (or algebraic integer) coefficients that are small enough (compared to the size of the coefficients) at the point (WI, . . . , w,). We already discussed this in $51.4 and 4.3 of Chapter 2. Suppose that the forms Lj = Li(Z)

= &jZj,

i=l,...,m,

(8)

j=l

are linearly independent, and maxi~j~m anya= (WI,... ,w,) E Cm one has

laijl < Hi. It is easy to see that for

> c1 PI ci=lm IJw’>l Hi H~*e*Hrn ’ where A = detllaijll and cr = &I maxr~j~m Iwjl. Suppose that for all N sufficiently large we are able to construct of linearly independent linear forms (8) with aij E Z such that

IL&ql = o@v2-“),

ma laij I L N, &j

Further suppose that the numbers i.e., there exists a linear form @)

= blzl

WI,. . . , w,

lFjym -- PA = H,

l
are linearly

+ . . . + bmzm,

a system

dependent

over Q,

bjE%,

HZl,

for which I(Z) = 0. We can choose m - 1 of the forms in (8), say L2, . . . , L,, so that they together with 1 form a complete set of linearly independent forms. If we apply (9) to this set of forms and take into account that IAl 2 1, we find that m c i=2

IL@)l

2 $v2-,

.

216

Chapter 5. Functions That Satisfy Linear Differential

Equations

But this contradicts (10). Hence, the existence of the forms (8) satisfying (10) implies linear independence of the numbers wi , . . . , w,,, . If one is able to construct a set of forms (8) with coefficients aij E Zx, where IK is an algebraic number field of degree [K : Q] = h 2 1, and with (10) replaced by maxi i,j

5 N,

IL@)l

= o(W-“),

l
(11)

where v > 0, then from (9) one can only derive a lower bound for the cardinality T of a maximal subset of the numbers wi, . . . , wm that are linearly independent over K. In that case there exist linearly independent linear forms ii(Z) = bilX1 + * * * + bi,Xmy

i=l

,...,m-T)

(12)

maxm = H, H21, Gj for which ii(W) = 0, i = 1,. . . , m - r. If we apply (9) to the forms (12) and a subset L,-,+I, , . . , L,,, of the forms (8) (which are linearly independent along with the forms 11,. . . , I,-,), we find that bij E

ZK,

2

ILi(Z)j

> &N1-TIAl

.

(13)

km--r+1

Since A is a nonzero algebraic integer, we have IAl 1 m account that [al < cgN’, and hence IAl > ~siV-‘(~-i), we obtain the inequality 1 - v > 1 - rh, i.e., T>K

-(‘+l). Taking into from (11) and (13)

h’

In the proof of the next lemma (Lemma 16 of Chapter 3 of [Shidlovskii 1989a]) we construct the linear forms (8) satisfying (11) that are needed to prove Theorem 5.5. Lemma 5.1. Suppose that the E-functions fi(z), . . . , fm(z) form a solution of the system of differential equations (3) and are linearly independent over C!(z). Let CYbe any algebraic number not equal to 0 or a singular point of the system (3), and let K be an algebraic number field that contains cr and the Taylor coeficients of the E-functions fi(z), . . . , fm(z), [K : Q] = h. Then for any E, 0 < E < 1, and for all n E N suficiently large there exist linearly independent linear forms (8) with aij E & such that maxl=)=O i,j

IJuf1(a),

. * * 7fm(a))l

( n (l+a)n), = 0 (n-(m-l-E)n)

i,j=l,..., ,

m, i = 1,. . . ,m .

(15)

(16)

Let T be the cardinality of a maximal subset of {fi(a), . . . , fm(a)} that is linearly independent over IK. It follows from Lemma 5.1 and (14) that

52. The Siegel-Shidlovskii m r ’ h(l+&) Since E is arbitrary, Corollary

5.1.

Method

217

.

we have the following Under the conditions in Lemma 5.1, r>m/h.

We now show how Theorem 5.5 follows from this corollary. Suppose that the hypothesis of Theorem 5.5 is satisfied, and the functions (5) are algebraically independent over C(z). Then for any N E N the products (4) are linearly independent over C(z). One easily verifies that these products form a solution of a system of homogeneous linear differential equations whose singular points are the same as those of the system (6). Since E-functions form a ring, all of the products in (4) are E-functions. If we apply Corollary 5.1 to them, we can conclude that among the numbers f~1(4--~f~m(4,

O
kl+...+k,
(17)

there are at least + 0 (N-l)

(18)

that are linearly independent over K. If the numbers fi (o), . . . , f,,,(o) were algebraically dependent over Q, then it is easy to see that the maximum number of numbers (17) that are linearly independent over lK would have order 0 (Nm-l), and this contradicts (18). This shows that the numbers in (7) are algebraically independent. The converse assertion in Theorem 5.5 is trivial. It would be very nice to prove the following generalization of Corollary 5.1. Conjecture 5.1. Suppose that the E-functions fi(z), . . . , fm(z) form a solution of the system of differential equations (3) and are linearly independent over C(z). Let cr be any algebraic number not equal to 0 or a singular point of (3). Then the numbers fi(a), . . . , fm(a) are linearly independent over the field of algebraic numbers A.

This conjecture holds in the special case when fi(z) = ecyiz and a! = 1 (Lindemann’s theorem). It is also true when lK = Q, since in that case h = 1 and Corollary 5.1 implies that r = m. Moreover, Conjecture 5.1 also holds when K is a quadratic imaginary field, despite the fact that h = 2. In fact, the algebraic integers of such a field form a lattice in Cc.Hence, the algebraic integer A in (13) satisfies the bound /Al 2 cs for some constant cs > 0 that depends only on K; it is easy to see that this implies that r > V, and we have r > m/(1 + E) in Lemma 5.1, i.e. (since E is arbitrary) r > m. For K an arbitrary algebraic number field, the conjecture has been proved under the additional assumption that

218

Chapter 5. Functions That Satisfy Linear Differential

Equations

tr deg,WI(Z), . . . , fm(z>> = tr degq,)W(z), . . . , fm(z>) (see Theorem 9 of $8, Chapter 4 in [Shidlovskii 1989a]). In [Nesterenko and Shidlovskii 19961 it is proved that Conjecture 5.1 holds for all (Y E A outside of a finite set. Finally, note that Conjecture 5.1 trivially implies Theorem 5.5. .2.2. Construction of a Complete Set of Linear Forms. A method for constructing a complete set of linearly independent linear forms satisfying (15) and (16) was suggested to Siegel by Thue’s proof of his theorem on approximating algebraic numbers by rational numbers. First, to obtain one such form, one constructs a functional linear form R(z)

=

Pl

(Z>fl(Z)

+

** * +

~m(z)fm(z)

,

(19)

Pi(z) E Z,[z], that has a zero at z = 0 of high order (ord R) compared to the degrees and magnitudes of the coefficients of the Pi(z). More precisely, for any E, 0 < B < 1, and for all n sufficiently large, there exist polynomials Pi(Z)

=

2bijZ’y

bij

E &I

i=l,...,m,

j=O

such that i=l,...,

1 = 0 (72’1+E)n) )

ma;trbij

m, j=O,l,...,

ordR 2 m(n + 1) - [en] - 1 .

n,

(20) (21)

The condition (21) means that the m(n + 1) unknown coefficients bij must satisfy m(n + 1) - [en] - 1 linear relations. It remains to prove that this system of homogeneous linear equations has a nontrivial solution satisfying (20). The function R(z) is an E-function. Hence, its Taylor coefficients approach zero very rapidly, and from (20) it immediately follows that for any E > 0 one has pqcr)I = 0 (n+--l--c)n) . Thus, we have constructed one linear form b(fl(Q),--*

,fm(a))

=

R(a)

=

s(a>fl(a)

+

**-

+Pm(~)fm(~)

in the numbers fi (a), . . . , fm(a) with algebraic coefficients that is rather small in absolute value. In order to construct a complete set of such forms, one first constructs a complete set of linearly independent functional linear forms h(z)

=

Pkl(Z)fl(Z)

+

***

+

Pkm(Z)fm(Z),

k=l,...,m,

(22)

that vanish to a high order at z = 0. If we differentiate both sidesof (19), then replace fi (z) , . . . , fA(z) by the expressions on the right in (3) with fi(Z) in place of yi, and finally multiply the result by a common denominator t(z) of

$2. The Siegel-Shidlovskii Method

219

the coefficients in (3)) then we obtain another linear form in fi (z), . . . , fm(z) with polynomial coefficients. The degrees of the coefficients do not increase much, and the multiplicity of zero at a = 0 of the new linear form is at most 1 less than that of R(z). This procedure can be repeated several times. In this way one constructs the linear forms (22): Rk(z) =

RI(Z) = R(z),

t(Z)-

d&--l dz

k=2,3,...

’

.

We now assume that the forms (22) turn out to satisfy the condition A(Z)

=

det

iipkiIli,k=l,...,

m

#

0 .

(23)

, Rm(a) will serve as the required set of linear If A(a) # 0, then RI(~),... forms in Lemma 5.1. If, on the other hand, A(a) = 0, it turns out that the polynomial A(Z) does not have a high order of zero at z = cr. Using bounds on the degrees of the &(z), one easily shows that degA(z)

5 mn + cl ,

where cl does not depend on n. Using Cramer’s formulas, it is also easy to prove that ord A(z) > mn - [En] - cz , and hence A(z) = zmn--jen]-cczA1(~) , where deg Al(z)

5 in + cs. But then there exists an integer 7, O
such that A(T)(a) # 0. It is not hard to seethat this implies that there are at least m linearly independent linear forms among the forms pkl

(a)21

+

’ ’ ’ +

Pkm(a)%,

l
These m forms will serve as the set of linear forms in Lemma 5.1. 2.3. Nonvanishing of the Functional Determinant. It remains for us to ensure that (23) holds. It was for this reason that Siegel in [1949] imposed his normality condition on the system (3). Definition 5.2. The system (3) is said to be normal if it can be divided into subsystems in such a way that every set of nontrivial solutions of a subsystem consists of functions that are linearly independent over (c(z). Our terminology is slightly different from Siegel’s in [1949]. Siegel used the term “normal” for a solution fi (z), . . . , fm(z) of (3) rather than for the system (3) itself. He said that such a solution is normal if fi (z) . . . fm(z) # 0 and the system (3) satisfies our definition above.

220

Chapter

5. Functions

That

Satisfy Linear

Differential

Equations

Example 5.1. If or,. . . , om E Ccare distinct, then the system I Yi = &Yi,

i=l,...,m,

is normal, since it splits into m subsystems, each consisting of a single differential equation, that satisfy the condition in Definition 5.2. Example 5.2. For X # 0 the system

1 Y;

=

(1

y;

=

0

-

$)

Yl

+

;Yz

,

is not normal, since its nontrivial solution yr = z-XeZ, ys = 0 has linearly dependent components, and the system cannot be split up into subsystems. Recall that to prove Theorem 5.4 one needs to apply Siegel’s normality condition to a system of differential equations that is satisfied by the products (4). It is easy to seethat, if the functions fr (z), . . . , f,,,(z) form a solution of the non-homogeneous system (6), then the normality condition will not generally hold for the system of differential equations satisfied by the corresponding functions (4). For example, in the case of the function

CPA(Z)= 1+ n=l fy

(A + 1) .“:: (A + n) ’

which satisfies the differential equation y'=

1-G (

y+I, >

Z

the set of functions (4) with N = 1 has the form yi = 1, ys = VA(Z). This pair of functions is a solution of the system (24), which, as we saw, is not normal. In 1953 Shidlovskii found a different sufficient condition, which he called irreducibility. If the system (3) satisfies this condition, then the inequality (23) holds. Definition 5.3. The system (3) is said to be irreducible if the nonzero components of any solution are linearly independent over C(z).

Actually, Shidlovskii used the term “irreducible” not for the system (3), but rather for the solution set fr(z), . . . , fm(z), and he imposed the additional condition that fr (z) . . . fm(z) # 0. It is easy to see that a normal system of differential equations satisfies the irreducibility condition. The converse is false. For example, if X E @\ Z, then the system (24) is irreducible, since any solution has the form yl = clcp~(z) +

c2zpxez,

Y2 = Cl ,

and one easily verifies that the condition in Definition 5.3 is satisfied.

$2. The Siegel-Shidlovskii

Method

221

The irreducibility condition is less restrictive than the normality condition. In particular, it enables one to extend Theorem 5.4 to functions that satisfy a non-homogeneous system of differential equations (see [Shidlovskii 1954, 1959131). In the same papers Shidlovskii also obtained several new results on the values of concrete E-functions. In [1955a, 1955b,1959a] Shidlovskii was finally able to prove (23) without any additional conditions on the system of differential equations (3). This enabled him to prove Theorem 5.5. It is useful to state the result of these papers in a form different from (23). The following lemma has the advantage that it is easier to generalize and refine. Lemma 5.2. Suppose that the functions fi(z), . . . , fm(z) form a solution of the system of differential equations (3), are analytic at the point z = 0, and are linearly independent over Cc; PI (z), . . . , P,(z) are polynomials in z of degree at most n; and R(z)

=

s

(Z)fl(Z)

+ . *. + P*(z)fm(z)

If the C( z )- vet t or space spanned by the functions dimension s, then ordR(z) 5 sn+cc, zuhere ord R(z) is the multiplicity of zero of R(z) is a constant that does not depend on n.

Rci)(z),

i = O,l, . . ., has

(25) at the point z = 0, and GJ

From Lemma 5.2 and the bound (21) with E < 1 and large n it follows that s > m - 1. But this means that s = m, and the functions and also the linear forms RI(Z), . . . , Rm(z) are linR(z), R’(z), . . . , R(m-l)(z) early independent over C(z), i.e., (23) holds. The operation of constructing new functional linear forms from the ones in (19) by differentiating and then replacing f:(z), . . . , f:,(z) by the expressions on the right in (3) can be interpreted as an action on linear forms R = PI(z)zI of the differential

+ ... + P,(z)z~

operator D=

The underlying reason for the inequality (25) is that any space L of linear forms over C(z) that is closed under the action of D has a basis consisting of linear forms whose coefficients are polynomials in z of degree less than a constant that does not depend on L. For example, when the system (3) is irreducible, any such space has a basis of the form xil, . . . , xir.. 2.4. Concluding Remarks. The method described above, which originated in the late 1940s and early 1950s in papers by Siegel and Shidlovskii, was subsequently developed and generalized much further. Shidlovskii himself wrote

222

Chapter 5. Functions That Satisfy Linear Differential Equations

a series of papers on algebraic independence of the values of E-functions in caseswhen there are algebraic relations among them, on quantitative results that generalize and refine Theorem 5.3, on methods of proving algebraic independence of the solutions of linear differential equations, and on applications of the general theorems to concrete Efunctions (see [Shidlovskii 1989a]). In recent years this area of research has attracted the attention of many specialists. The sections that follow will describe some of the results that have been obtained in the course of developing the Siegel-Shidlovskii method.

$3. Algebraic Independence of the Values of Hypergeometric E-Functions Most of the results in this section are consequencesof Theorem 5.5. Because of that theorem, the results follow once one establishes that the corresponding E-functions are algebraically independent. The conditions on the parameters are there essentially for the purpose of ensuring algebraic independence of the E-functions over c( 2). 3.1. The Values of Efunctions That Satisfy First, Second, and Third Order Differential Equations. The simplest hypergeometric Efunction - the case m = 1 - was mentioned in §2: 03

CPA(Z)

=

1Fl

x+1;zl (

= >

c (A + 1) .“:: (X + n) . n=O

Note that cps(z) = e’. The next theorem, which was proved by Shidlovskii in [1959c], generalizes the Lindemann-Weierstrass theorem. Theorem 5.6. Suppose that Xo E N, XI,. . . ,A, E Q \ Z, and none of the diflerences Xi - Xj are integers, 1 5 i < j 5 n. Further suppose that , pm E A are linearly independent over Q; and (~1,. . . , (Y,,, E A are Pl,... distinct and nonzero. Then the (n + 1)m numbers cPXO(Pi)7

CpXj

Cai>>

llilrn,

11jSn,

are algebraically independent over Q. In view of (2), the conditions in Theorem 5.6 are necessary as well as sufficient. Note that to prove this theorem one must show that the E-functions cPXO(Pi’)> ( (Yi z > are algebraically independent over e(z). A similar technique is used to prove other theorems on the values of E-functions at various points. In the case m = 2 one has two hypergeometric Efunctions: CpXj

53. Values of Hypergeometric Kx,&>

=

lF2

1 X+l,/J++C

In [1959b] Shidlovskii

EFunctions

223

z2 &w(z)

> ’

proved the following

l,u+l x+l,/J+P

= 2F2

generalization

of Theorem

> . 5.1.

Theorem 5.7. Suppose that X,p E Q, and X - p is not equal to half of an odd number. Let CYbe any nonzero algebraic number. Then Kx,+(a) and K:,,(o) are algebraically independent over Q.

This theorem was strengthened by Belogrivov [1970] and Oleinikov [1970], who, using different methods, obtained a definitive result. Theorem 5.8. Suppose that X,p E Q are not in No = (0, 1,2,. . .}, and Q is a nonzero algebraic number. The numbers K~+(cY) and Ki,,(a) are a1gebraically dependent over Q if and only if 2X and 2,3 are integers of different parities (i.e., one odd and one even).

One also has the following generalization of Theorem 5.2 (see [Shmelev 19691and [Nesterenko 19691). Theorem

5.9. Suppose that X1,. . . ,X,,

~1,. . . , pLmE Q \ No, and

for ICI, ka E Z. Further suppose that cf,. . . , ri are distinct nonzero algebraic numbers. Then the 2mn numbers Kxw

(SjL

Kii,& (O),

i=l,...,

m, j=l,...,

n,

are algebraically independent. As early as 1929, Siegel pointed out the possibility of studying the values of the Kummer function A~,o,+(z), which satisfies a second order homogeneous linear differential equation. The first partial results were proved in [Yakabe 19591. In [1962] Oleinikov proved the following theorem. Theorem

5.10.

Let X and u be rational numbers with

u 4 %,

u-X$N,

A# -l,-2,...

.

If < is a nonzero algebraic number, then the numbers A,Q~,“(<) and A’,,,,,(c) are algebraically independent. In 1966 Shidlovskii found necessary and sufficient conditions for AA,~,“(~) and A’x,P,v(<) to be algebraically independent. Theorem

5.11.

Suppose that X, ~1,v E Q are not equal to negative integers,

and u-X$N,

u-pq!N.

If 5 is a nonzero algebraic number, then the numbers AX,+“(<) are algebraically independent.

and A’,,,,,(<)

224

Chapter 5. Functions That Satisfy Linear Differential

Equations

In [1971] Belogrivov proved an analogue of Theorem 5.2 for the Kummer function. The proof of this theorem led to results on the general case as well. Theorem 5.12. Suppose that Xj, pj, uj E Q, j = 1,. . . , m, are not equal to negative integers, and suppose that none of the differences

-

Uj

(vj

-

Xj>

"j

Xj,

- (Q - Xk),

(vj

-

j # k,

Xj)

-

Pj,

11j<m,

- (Q - ilk),

("j

-

Pj)

- (WC- P/c) Y

llj,klm,

are in Z. If the algebraic numbers 51,. “, [ n are linearly independent over Q, then the 2mn numbers

are algebraically independent. A proof of this theorem was published by Gorelov [1981], who also proved a theorem giving sufficient conditions for algebraic independence of the values of the functions Kx+(z), A+,, (z) and their derivatives at different points for various values of the parameters. Earlier (see [Belogrivov 19711) a similar theorem had been proved for the values of the functions Kx+(z), Ax,o+,(z), eI . In [1968] Oleinikov studied the following hypergeometric E-functions, which satisfy third order differential equations:

For each of these functions he found sufficient conditions for the values of the function and its first two derivatives to be algebraically independent. In [1969] he obtained the following improved version of this result. Theorem

5.13.

Suppose that X and p are rational numbers not equal to

negative integers,

f(z)=

1F3

( 1,x+ l,p+ ’

’ 3 ,

1; (03

and (Y is a nonzero algebraic number. The numbers f(o), f’(a), and f”(a) are algebraically dependent if and only if 3X, 3~ E Z, and either 3X E 1 (mod 3), 3p E 2 (mod3), or else 3X - 2 (mod 3), 3~ = 1 (mod 3). In [1968] Oleinikov cited a result (Lemma 3) that later turned out to be incorrect, because it had been stated in too much generality. The proofs of the theorems concerning r Fs and 3F3 could easily be patched up; in particular,

$3. Values of Hypergeometric

E-Functions

225

Theorem 5.13, which only needs them for 1Fs, is still correct. The result for 2F3 in [Oleinikov 19681 is also valid, but the proof requires more substantial modifications (due to Salikhov). In any case, it should be noted that all of these results are consequences of general theorems that were later proved by Salikhov (see Theorems 5.14, 5.16, and 5.17 below). 3.2. The Values of Solutions of Differential Equations of Arbitrary Order. Based on ideas of Oleinikov, in [1977] Salikhov found sufficient .conditions for algebraic independence of the values of the function x1 + 1

cp(z) = 1%

1

(26)

,‘-‘I

and its successive derivatives through the (m - 1)-th order, where m is any prime number. Using the results in [Salikov 19771, Oleinikov [1986] found some other sufficient conditions. Finally, in [1988a] Salikhov proved the following generalization of Theorems 5.8 and 5.13. Theorem 5.14. Suppose that m is a prime number, Xi,. . . , A,,, E Q are not equal to negative integers, and [ is a nonzero algebraic number. The numbers

are algebraically dependent over Q if and only if the numbers mX1, . . . , mX, are integers and form a complete system of residues modulo m. In [1989, 199Oa] Salikhov was able to prove this assertion for odd m, and even extend the result to the values of p(z) at different points. Theorem 5.15. Let q(z) be given by (26), where m is an odd integer and Xi # -1, -2,. . . . Suppose that [I,. . . ,& are nonzero algebraic numbers such that
O
lSj
algebraically dependent over Q if and only if the numbers mAi,. integers and form a complete system of residues modulo m.

. . , mX,

In [1990] Cherepnev was able to replace the condition
=

~+lFm

h,...,LI

t=m-l>O,

(27)

where in what follows ~1,. . . , PI, Xi,. . . , A, are assumed to be nonzero rational numbers that are not equal to negative integers. It is convenient to use the following notation:

226 1) 2)

Chapter 5. Functions That Satisfy Linear Differential

Equations

for a = (al,. . . ) a,) E @” and c E C we let (E + c) denote the vector (al+c,...,a,+c); a,) E Cc’ and 6 = (bl,. . . , b,) E Cc’, then we write z FS 6 if B = (al,..., if there exists a permutation il,. . . ,i, of the indices 1,. . . ,s such that - bij E Z for j = 1,. . . ,S. Uj

Theorem 5.16. Let g(z) be given by (27). Suppose that t = m - 1 is odd, 2 > o;x= (Xl )...) X,),p= (p1, . . . , PI), and t is a nonzero algebraic number. Further suppose that the following conditions are fulfilled: 1) 2)

Xi - pj $ Z for i = l,...,m, j = l,...,l; there is no integer d > 1 that divides both t and 1 and has the property that ix+;, x,x, (p+i) wp.

Then the numbers +(a are algebraically

#a..

.7 P+“(o

independent.

When = m - 1 is even, the conditions become much more complicated. For a proof of the next theorem, see [Salikhov 1989, 1990b]. t

Theorem 5.17. Suppose that t = 27 is even, 1 1 0, r + 1 > 2, and 1 > 3 if t = 6. Further suppose that conditions 1) and 2) in Theorem 5.16 are fulfilled, along with the following four conditions: a)

if

1 = 0, then there do not exist rational which (X+x0)

b)

c)

condition

(X+x0)

M (0,1/2,zl,-21)...,

@+x0)

=

(

b) holds with

(Ti

+

$0)

-

(21,-Xl,...,

=

(0,

l/2,

(p

+

If < is a nonzero

x0)

. . ,x,-l

for

x7+,,

-G+s,

then there do not exist rational

XT+,-1, . * * ,x,+&l,

-&+a-1

> >

-x,+1-1)

(28)

;

(28) replaced by: XT+87 G+s+1,

-x7+8

> 9

--z,+s+1,

* . . , x,+1---1

if 1 = 2s + 1 is odd, then there do not exist rational ***, x,+1-1 for which (~+xo>

x0,x1,.

(~,~/~,~l,-~l,...,~~-l,-~s-l);

if 1 = 2s > 0 is a positive even number, numbers x0, x1, . . . , x,+1--1 for which

(X+x0)

d)

=

numbers

=

(0,x1,

-x1,.

. . ) &+s,

-

(l/2,

G+s+1,

-x,+8+1,

algebraic number,

-x,+s)

numbers

;

x0, xl,.

)

* * * , &+I-1,

then the numbers

7 -x,+1-1)

-2,+1-d

.

..

$3. Values of Hypergeometric

$a

E-Functions

227

Pw), . . . 3ti’“-YE)

are algebraically independent. In [Beukers, Brownawell, and Heckman 19881a general theorem was proved about algebraic independence of the values of hypergeometric Efunctions with different sets of parameters. Following these authors, we shall denote fs(z)=~F~-l

(,p”*‘;“~ ,"',

;(-z)~), m

t=m-l>O,

1

where for brevity

we let S denote the set of real number parameters S = , A,}. Here we assumethat m > 12 0, m 2 2, and A, = 1. We say that S is an admissible set of parameters if it satisfies at least one of the following conditions: bl,..

a) b)

.,pLI,h,--.

Xj - pi $! Z for 1 5 i 5 1, 1 5 j 5 m, and all of the sums Xj + pi for 1 5 i 5 j 5 m are distinct modulo Z; 1 = 0, m is odd or equal to 2, and the set {Xi,. . . , A,} modulo Z is not a union of arithmetic progressions {A, X + i, . . . , X + y} of fixed length d, where dim, d > 1.

WesaythattwosetsofparametersS={C11,...,~I,X1,...,Xm}andS’= {pi,. . . ,p;,,Xl,,. . . ,A;,} are similar if a) b)

1’ = 1, m’ = m; there exist ~1,X E lR and r E (0, 1) such that with a suitable indexing of the parameters one has pi E /.L+ (-l)‘pi A> z X + (-l)‘Xj

(mod Z), (mod Z),

i = l,...,l, j=l,...,m.

The next theorem is proved in [Beukers, Brownawell, and Heckman 19881. Theorem 5.18. Let S1, . . . , Sk be admissible sets of rational parameters (not necessarily all distinct), and let &, . . . , tk be nonzero algebraic numbers such that if Si and Sj, i # j, are similar sets of length 1,m, then *
f&&k), f&k,, 0.‘~ @-%k) , (where mi is the value of m for Si) are algebraically independent over Q. In particular, this theorem contains Siegel’s Theorem 5.2. It is interesting to compare Theorem 5.18 with Salikhov’s Theorem 5.15, where we take

228

Chapter

5. Functions

That

Satisfy Linear

Differential

Equations

A, = 1. We shall also suppose that 1 = 0 and m is odd, i.e., condition b) in the definition of admissibility is fulfilled. If we ignore the numbers ~1, . . . , n8 in Theorem 5.18, then we can say that Theorem 5.15 imposes stricter conditions on the points
5.19.

Let (Y be a nonzero algebraic number, and let

wk(z)=l+g+& n=l

lC>l.

.

Then for any r E N the numbers ea, WI(a), . . . , w,(o) pendent over Q. Theorem

are algebraically inde-

5.20. Let a be a nonzero algebraic number, and let W&k(Z)

= 2 Zn n=. n!(n + X)k ’

k>l.

Then for any X E Q, X # 0, -1, -2,. . ., and for any r E N the numbers e”,wx,l(a), . . . , WA,,.(~) are algebraically independent over Q. Theorem 5.20 implies, in particular, that for any nonzero Q!E A the numbers e”’ and Jo* eet dt are algebraically independent, as are the numbers e” and JoaP-l emtdt. Theorem

5.21.

Let

Then for any nonzero (Y E A and any r E N the r(r + 1)/2 numbers tip cay> 3

O
l
are algebraically independent. The next result is proved in [Belogrivov 1967, 19701. Theorem

and

5.22.

Suppose that A, ~1E Q, A, /I # 0, -1, -2,. . .; ml, ms E N;

$4. The Values of Algebraically

Dependent E-hnctions

229

m=ml+mz,

If

< is a nonzero

algebraic

number,

if

and

A(S),A’(t), . . . , A”-(5) are algebraically dependent over Q, then one of the numbers X or p is an integer, and the other one is half of an odd number. Note that when ml = m2 = 1, this theorem contains Theorem 5.8. In [1967] Belogrivov also proves a result about functions that have the same form as A(z), but with more than two expressions (X)ri in the denominator of the Taylor coefficient. In [1968b] Mahler proved several theorems about algebraic independence of the values of concrete Efunctions. In particular, he proved that for any m E N, nonzero X E Q, and nonzero { E A the numbers

J txml 1

In” t e@dt,

Ic=O,l,...,

m,

0

are algebraically independent. (Shidlovskii later showed that es can be added to this list of algebraically independent numbers.) The exact statements and the proofs of these and many other results can be found in the books [Mahler 19761and [Shidlovskii 1989a].

94. The Values of Algebraically

Dependent

E-Functions

Suppose that each of the functions fi (z), . . . , fm(z) has a Taylor expansion near z = 0 with coefficients in A. If these functions are algebraically dependent over C(z), then it is easy to see that there is a polynomial P with algebraic coefficients such that P(z, fi,. . . , fm) = 0. We may assume that P is an irreducible polynomial, so that, given any cx E A where fi (z), . . . , fm(z) are defined, the polynomial P(Q,z~, . . . ,z,) E A[zl, . . . ,z,] gives a nontrivial algebraic relation over A among the numbers fi(a), . . . , fm(a). This implies that fi(cr),... , fm(a) are algebraically dependent over Q. By repeating this argument, we can show that tr degQ Q(fl(a),

. . . , fm(a)) 5 tr degqZ) c(z, fl(z),

. - . , fm(z)) .

(29)

From the example fl(Z)

=

z,

f2(z)

=

(z

-

l)e’,

cK=l,

we seethat it is possible to have strict inequality in (29). To prove that equality holds in (29) one needs to impose restrictions on the functions fi(z) and the point a.

230

Chapter 5. Fnnctions That Satisfy Linear Differential

In this chapter, as before, . . . , fm(z) are Efunctions.

we shall assume

4.1. Theorem on Equality proved the following theorem ‘Theorem

that the functions

fi(z),

...

of ‘kmscendence Degree. In 1956 Shidlovskii (see Chapter 4, $4 of [Shidlovskii 1989a]).

5.23. Suppose that the E-finctions h(z),..-,fm(z),

form

Equations

a solution

(36)

m21,

of the system of m linear difierential

y; = Qm(z) + 2

k=1,2

Qdz)~i,

equations ,...,

m,

(31)

i=l

where Qki(z) E C(z), of any of the Qki(z). tr dego Q(fl(a),

and cr is any algebraic number not equal to 0 or a pole Then . . . , fm(a))

= tr degq,)

c(z, fl(z),

. . . , fm(z))

.

This is a generalization of Theorem 5.5 (Shidlovskii’s First Fundamental Theorem). The example given above shows that the theorem can fail if cr is a singular point of the system. This theorem implies, in particular, that sin (Y and cos o are transcendental for any nonzero algebraic number o. This means that tan o is also transcendental. The basic technical result needed to prove Theorem 5.23 is the following lemma (see Lemma 2 of Chapter 4 of [Shidlovskii 1989a]). Lemma 5.3. Suppose that the functions fi(z), . . . , fm(z) the homogeneous system of differential equations

YL= 2

Qdz)~i,

k=

1,2 ,...,

m,

Qki(z)

form

E W)

a solution

of

,

i=l

and the maximal number of functions fi(z) that are C(z) is equal to s. Let Q be any complex number not of the system. Then one can choose s of the functions linearly independent over C(z) and satisfy a system ylc = f= Q;i(z)yi,

k=1,2

,...,

s,

linearly independent over equal to a singular point fi (z), . . . , fm(z) that are of differential equations

&L(z)

E W>

>

i=l

for which z = CY is not a singular Using this lemma and Corollary

point. 5.1, one obtains

Proposition 5.1. Suppose that the E-functions fi(z), . . . , fm(z) form a solution of the system of differential equations (3); (Y is any algebraic number

$4. The Values of Algebraically

Dependent EF’unctions

231

not equal to 0 or a singular point of (3); and K is an algebraic number field that contains (Y and the Taylor coeficients of the E-functions fl(z), . . . , f,,,(z), [K : Q] = h. Let s be the maximal number of fvnctions fi(z) that are linearly independent over C(z), and let r be the maximal number of numbers fi(a) that are linearly independent over Q. Then r > s/h. Theorem 5.23 easily follows from this proposition. Let u=trdegQ~(fl(a),...,fm(a)); 21= tr degqz) W,

fl(z),

. . . , fm(z)) ;

and let cp,,(N) and cpv(N) denote the dimension of the K-vector space spanned by O
space spanned by

O
f;r(z),...,f~m(z),

k1+***+lc,
respectively. (The functions v,(N) and v,(N) are called the “characteristic function of the ideal of algebraic relations among fl(ct), . . . , fm(a) over IIS’ and the “characteristic function of the ideal of algebraic relations among fl(Z)Y. . . , fm (z) over C(z),” respectively.) By Proposition 5.1, for all N > 1 we have cpu(W 2 $m)~

(32)

For N sufficiently large, p,(N) and v,(N) are polynomials in N of degree u and 2r, respectively (see the Hilbert-Serre Theorem on p. 232 of Vol. 2 of [Zariski and Samuel 19601). Then (32) clearly implies that u 2 w. This, along with (29), gives Theorem 5.23. 4.2. Exceptional Points. Although it is a quite general theorem, Theorem 5.23 leaves some questions unresolved. For example, it does not give us transcendence of the value of one of the concrete E-functions in (30). Actually, such a value is not necessarily transcendental. Examples show that even at a nonsingular point of the system of differential equations it might happen that a subset of the functions (30) that are algebraically independent over C(z) take values that are algebraically dependent over Q. For instance, the E-functions fi(z) = (z - 1) e+ and f~(z) = ze’ are algebraically dependent over C(z) and form a solution of the system of differential equations

{

Yi = Y2 > Y& = -y1 + 2Y2 .

At the nonsingular point z = 1 the transcendental function fi(z) takes an algebraic value. This example also shows that the condition Q # 0 in Theorem 5.23 is essential, since both fi(z) and fi(z) take algebraic values at z = 0, i.e., for (Y = 0 one no longer has the equality of transcendence degrees.

232

Chapter 5. Functions That Satisfy Linear Differential

We see that there sometimes

exist so-called

exceptional

Equations points

for a subset (33)

fl(Z),...,fk(Z)

of the functions (30) that are algebraically independent over Cc(z). An algebraic point that is not equal to 0 or a singular point of the system of differential equations is said to be an “exceptional point” if the functions in (33) take values there that are algebraically dependent over Q. In several of his papers Shidlovskii studied the question of when such points arise and how to find them. In the first place, the set of exceptional points is always finite (see Theorem 5 of $6, Chapter 4 in [Shidlovskii 1989a]). In particular, we can say that, under the conditions in Theorem 5.23, every transcendental E-function in (30) takes transcendental values at all algebraic points with only finitely many exceptions. Under the same assumptions, almost all of the zeros of such a function (i.e., all but finitely many) are transcendental. The same can be said about the points where such a function takes algebraic values. The next two theorems (see [Shidlovskii 1989c, 19921) show that the exceptional points come from the coefficients of the algebraic relations over C(Z) among the functions (30). Theorem 5.24. Suppose that the E-functions (30) form a solution of the system of differential equations (31) and have power series coeficients in a quadratic imaginary field IK. Further suppose that tr degq,) C(fi(z), . . . . * * 7fm(z)) = k, and the functions (33) are algebraically independent over C(z). Let 01E K be nonzero and not equal to any of the singular points of the system (31). Then the numbers

are algebraically dependent over Q if and only if there exists a primitive irreducible polynomial P E K[z][xl, . . . ,x,1 with P(fl(z), . . . 7fnz(z)) = 0 such that CYis a zero of all of the coeficients in P of monomials xi’ . . . x2 in which at least one of the variables xk+l, . . . , x, occurs. One has a weaker result when K is an arbitrary

algebraic number field.

Theorem 5.25. Suppose that the conditions of Theorem 5.24 are fulfilled, except that lK is an arbitrary algebraic number field of finite degree over Q. Then the numbers fl(~),...,fk(Q)

are algebraically dependent if and only if there exists an irreducible polynomial PEK[Z,Xl,... ,x~] with P(z, fl(z), . . . , fm(z)) = 0 such that P(cY,Zl,. . . ,x,)=pl(Xl,...,Xk)pZ(xl,...,Xn) and

Pl(fl(a),

. . . ,fk(a!)>

=

0.

in

~[~1~~~~7%]

$4. The Values of Algebraically

Dependent

E-Functions

233

The necessary and sufficient conditions in these two theorems shed light on the reasons why dependence relations arise among the values of algebraically independent functions, but they do not help us actually find the exceptional points. If certain additional conditions are fulfilled, there might not be any exceptional points at all. For instance, this is the case if we have tr deg, WI(Z),

. . . , fm(z>) = tr degq,) W(z),

. . . , fm(z))

(see Theorem 8 of 58, Chapter 4 in [Shidlovskii 1989a]). In some cases one can find a finite set in which all of the exceptional points must lie. For example, suppose that the functions (33) are algebraically independent over C(z), and there exists a polynomial B(z) E A[z] such that each of the functions B(z)fi(z), i = Ic + 1,. . . , m, is an algebraic integer over the ring A[z, fi(z), . . . , fk(z)]. Then all of the exceptional points are zeros of B(z) (see Theorem 6 of $7, Chapter 4 in [Shidlovskii 1989a]). Suppose that lK is a quadratic imaginary field and the functions fr (z), . . . . . . , fm-l (z) are algebraically independent, i.e., the functions in (30) are connected by an algebraic relation P(fr(z), . . . , fm(z)) = 0 that is unique up to a constant factor, where P(zi, . . . ,z,) is a primitive irreducible polynomial with coefficients in C[,Z]. In that case all of the exceptional points are common zeros of the coefficients in P of terms z{’ . . . xj,- with j, 2 1 (see Theorem 11 of $9, Chapter 4 in [Shidlovskii 1989a]). This result also follows from Theorem 5.23. If lK is a quadratic imaginary field, then one can find a certain finite set that must contain all exceptional points (see [Chirskii 1973a]). To do this one must construct a special basis PI,. . . , PN E A[z, xi,. . . ,x,1 of the ideal of algebraic relations among the functions (30), consisting of so-called minimal equations. One then writes out the coefficients of the leading terms of the Pi (where the monomials XT ... xj,- are arranged in lexicographical order). All of the exceptional points are zeros of these coefficients. It should be noted that in the general case Shidlovskii found an algorithm that enables one to determine a finite set that contains all of the exceptional points. Similar results have been proved for homogeneous algebraic dependence under the assumption that the E-functions under consideration form a solution of a homogeneous system of linear differential equations with coefficients in C(z) (see Chapter 4 of [Shidlovskii 1989a]).

234

Chapter 5. Functions That Satisfy Linear Differential

Equations

$5. Bounds for Linear Forms and Polynomials in the Values of E-Functions As early as 1929, Siegel wrote that his method would lead to lower bounds for linear forms or polynomials with integer coefficients in the values of Efunctions at algebraic points. To illustrate this possibility, he proved Theorem 5.3, which gives a bound for the algebraic independence measure of the values of the Bessel function Jo(z) and its derivative J;(z) at a nonzero algebraic point Q. In the same paper Siegel announced the result that for all integers bs, bi, bp, not all zero, and for any s > 0 one has

Ibo+ h Jo(t) + where

the constant

b2

J;(S)1 > cH-~-‘,

H = maxi&l

,

c > 0 depends only on E and < E Q.

5.1. Bounds for Linear Forms in the Values of E-Functions. We first consider the general situation. We have an m-tuple WI,. . . ,w,, and we want to find a lower bound for Il(W)j, where

lyjym -- PA = H,

H>l,

that depends on the parameter H. In order to show how Siegel’s idea works, we recall the method of proving linear independence that was given in 52. As in $2, we suppose that for some infinite and rather dense sequenceof natural numbers N we have been able to construct a set of linearly independent linear forms Li = Li(l)

= ?aijZjf

i=l,...,m,

(34)

j=l

oij E Z, for which ma laij I 5 N, i,j

ILi(GT)l = o(N~-“),

l
(35)

As in 82, we can choose m - 1 of the forms (34) - say, La,. . . , L, - so that they along with 1 give a complete system of linearly independent linear forms. If we apply the inequality (9) to this set of forms and use the fact that the determinant of the system 1, L2, . . . , L, is at least 1, we find that H-lIl(~)l

> c~H-~N~-~

- N-l

2

ILi(c)l

2 c~H-~N~+ If we now choose N large enough so that

- o(N1-“)

> (36)

i=2

.

$5. Forms and Polynomials in the Values of EFunctions

o(N’-*) < &H-‘N’-”

)

235

(37)

then from (36) we obtain the bound

Clearly, the smaller we choose N satisfying (37), the better the bound we obtain for ]1(a) 1.The optimal choice of N depends on H, and so the expression on the right in (38) ultimately is a function of H. This gives the required estimate. Once Shidlovskii improved and perfected Siegel’s method for studying the values of E-functions, it became possible to prove quantitative results in the general case as well. In the situation in Lemma 5.1, if we make the additional assumption that lK = Q, we obtain a set of forms Lo with aij E Z and max ]aij ] < c27~(‘+‘)~, id ILi(f~(a),

. . . , fm(a))I

<

c3n

i,j = l)...)

m,

(39)

7

i=l,...,m.

(40)

-(m-1--e)n

Here and later, the letter c with various subscripts denotes positive constants that depend only on (Y, E, and the Efunctions. If we choose N = C2n(1+E)n , [ we easily verify that (37) takes the form

1

mc3n-(“---E)n < clH-l

C2n(1+E)n 2--m> >

or H < C4n(l--E(“--l))~ .

(41)

In addition, if c is small enough so that 1 - .s(m - 1) > 0, and if n is the smallest natural number satisfying (41), then N w c5H(‘+“)/(1-“(“-1))

as

H+CO,

and (38) takes the form I@)[ > c,H-(“-l)(l+E)/(l-E(m-l))

.

(42)

By choosing E smaller and smaller, we can make the exponent of H on the right in (42) arbitrarily close to the best possible value 1 - m. A similar argument can be used in the casewhen K is a quadratic imaginary field. In [1967b] Shidlovskii proved the following general theorem. Theorem 5.26. Suppose that the E-functions fl(~),...,fm(Z),

m21,

236

Chapter 5. Functions That Satisfy Linear Differential

Equations

have power series coeficients in a quadratic imaginary field K, are linearly independent over C(Z), and form a solution of the system of differential equations Y; = 2

Qdz)~i,

k=1,2

,...,

m,

i=l

where Qki(z) E Cc(z). Let cx E IK be nonzero and not equal to a pole of any of the Qki(Z). Then for any E > 0 there exists a constant cr such that for bj E &, 1 5 j 5 m, satisfying

one has lhfi(a)

+ . . . + bmfm(a)l > c~H’-~-~

.

Of course, it is not yet possible to obtain such an estimate for an arbitrary algebraic number field K, since we cannot even prove linear independence of the fi(a) in that case. For hypergeometric E-functions the bound O(nEn) in parts 2) and 3) of the definition of an E-function can be replaced by O(P), where c is a sufficiently large constant depending on the ai and bj. If we have E-functions that satisfy this stronger condition, the estimates (39) and (40) can be refined. The term o(N2-“) in (35) becomes a more rapidly decreasing function, and this leads to an improved bound for the linear form. More precisely;the following theorem was proved in [Shidlovskii 1967a] (see also [Shidlovskii 19791). Theorem 5.27. Suppose that the conditions in Theorem 5.26 are fulfilled, and the E-functions satisfy the above stronger version of the definition. Then there exists a constant cs such that for any bj E Z, 1 5 j 5 m, satisfying

one has

lhfi(~) + -.++

b,f,(cr)I

> csH 1 - n--f~d(lnlnH)-“*

.

Here and later, the letter y with various subscripts denotes positive constants that depend only on (Y, K, and the constants in the bounds for the power series coefficients of the fj(z) in parts 2) and 3) of the modified definition of an E-function. For some functions one is able to construct a system of linear forms L,(Z) with even better simultaneous approximations and thereby obtain more precise bounds. This is usually done using an explicit construction of approximating forms. For example, if 5 is a nonzero point in a quadratic imaginary field I[, we have the following bound for linear forms in the values of the Bessel function, which improves upon the result announced by Siegel: for bi E i%n

55. Forms and Polynomials Ibo

+hJo(t)

+

baJ;(t)(

>

cgH-

in the Values of E-Functions 2

-

-&lInH)-'

,

237

H = max ]bi] > 0 .

In certain casesit is possible in this manner to obtain best possible estimates; such results are discussed in 54 of Chapter 2. 5.2. Bounds for the Algebraic Independence Measure. The bounds for linear forms in the last section can be used to obtain bounds for the algebraic independence measure of values of E-functions at algebraic points. General results of this type were first obtained in [Shidlovskii 1967133.In fact, any polynomial P(fi(o), . . . , fm(o)), where P E Z[zi,. . . ,z,], may be regarded as a linear form in the products of powers fi(a)“l . . . fm(cr)km, kl + . . . + k, 5 d = deg P. Since the set of E-functions form a ring, it follows that the products h(z) “‘.-.f,(~)~~,

kl+...+k,

Id

(43)

are Efunctions, and they satisfy a system of linear differential equations with coefficients in C(z). Thus, Theorem 5.26 can be applied to the functions (43), so as to obtain a bound for the algebraic independence measure of , fm(cr). A similar result can be proved in the more general situation fl(~y>,... when the fi(z) have power series coefficients in a quadratic imaginary field. Theorem 5.28. Suppose that the E-functions

.*,fnz(z),

h(z),.

ml17

have power series coeficients in a quadratic imaginary field K, are algebraically independent over C(z), and form a solution of the system of differential equations Y; = &o(z)

+ 2

Qki(z)Yi,

k=1,2

,...,

m,

i=l

where Qki(z) E C(z). Further suppose that (Y E K is not equal to 0 or a pole of any of the Qki(Z). Let d be a natural number. Then for any E > 0 there exists a constant u depending on E, d, (Y, and the functions fi(z)p such that for any polynomial P E Z[xl, . . . ,x,1 with P $0 and deg P 5 d one has

IP(fl (a>, . . . , f m (a))1 2 cH(P)‘-~-~

.

We note that this lower bound differs from the known upper bounds only by the E in the exponent of H(P). There is also an analogue of Theorem 5.27 for polynomials (see Chapter 13 of [Shidlovskii 1989a]). It is also possible to prove bounds for the algebraic independence measure of the values of E-functions in the case when the Taylor coefficients or (Ylie in an arbitrary algebraic number field of finite degree over Q. As early as 1929, Siegel proved the first such result for the function JO(Z) and its derivative

238

Chapter 5. Functions That Satisfy Linear Differential

Equations

(see Theorem 5.3). After Shidlovskii proved his First Fundamental Theorem (Theorem 5.5), it became possible to obtain a quantitative generalization of Theorem 5.3. This was done in [Lang 19621. Theorem

5.29.

Suppose that the E-functions h(z),

* * * 7 fm(z>,

m21,

have power series coeficients in an algebraic number field K of finite degree, are algebraically independent over C(z), and form a solution of the system of diaerential equations Y; =

Qko(z) + 2 Qki(z)~i,

k=1,2

,...,

m,

(44

i=l

where Qki(z) E C(z). Suppose that CI E IK is not equal to 0 or a pole of any of the Qki(z). Then for any P E Z[zl, . . . ,zm], P $0, one has IWl

(a>,

* * * 7 fm(a))l

L

cfwrbd”

7

where d = deg P, and b and c are positive constants, where b depends only on h = [IK : Q] and m, and c depends on m, cr, d, and the functions fi(z). In [Lang 19621this theorem was proved only in the case of a homogeneous system of differential equations. However, the proof in the general case does not require any important changes. In [1968] Galochkin obtained a refinement of Theorem 5.29 with the constant b computed explicitly. If the system of differential equations can be split up into r disjoint subsystems of orders ml,. . . , m,, then b = 2(2r)mmy1 . . .rnrr hm+i ml!. . em,! In addition, the term d” in the exponent of H(P) can be replaced by where dk is the total degree of P in the set of variables corre6;“’ ...drr, sponding to the functions fi(z) that form a solution to the k-th subsystem of differential equations. In particular, under the conditions of Theorem 5.29 we have b = 2”+‘mm(m!)-’ hm+‘. We briefly describe the basic ideas in the proof of Theorem 5.29. Suppose that we are given a set of linearly independent linear forms ii(z) = bilzl + . . . + bimzm,

bij E Z,

i=l,...,s,

andapointSj= (w~,...,w~) c(Cm, and we want to obtain a lower bound for in terms of the parameter H. Further suppose that for some m=l
$5. Forms and Polynomials

m=bijI id

5 N,

in the Values of ELFunctions

IL$.J)l = o(Nl-“),

lSi<m,

239

u>o.

Then, repeating the argument in 55.1 that led to (36) with appropriate modifications (see also §2.1), we obtain

H-’

l’=iy. I&(~)1 L cl (HW m--a)-h - N-l -2 cl (HsiVm-e)-h

2

IL&q

- o(iV-“)

2 (45)

i=s+1

,

from which we obtain an estimate for maxi 0. It is not hard to see that n can be chosen in such a way that

for some ~1 > 0 depending on E, and then from (45) we can obtain the following bound: , mu Ih(fi(a), . . . , fm(a))l > cZH1--h-=+ (46) l 0 and for some cs > 0 depending on Q, m, 6, and the functions fi(z). This bound is the same as (42) if s = h = 1. To prove Theorem 5.29 one applies the above considerations to the set of functions h(z) kl--f&)km,

k~+...+k,

5 (l+X)d,

(47)

where d = deg P and X is a large integer constant depending only on h = [IK : Q] and m. Then in the earlier argument the number of functions is replaced by the number

(

(1 + X)d+ m = (1 + X)m m, d” +0(&-l) m >

All of the polynomials Xlkl “‘2, km.P(sl

,...,

x,),

kl+.~.+k,<Xd,

may be regarded as linear forms with integer coefficients in the products of powers xf1 . . . xk) kl +.a. + k, < (1 + X)d , and one easily checks that these forms are linearly independent. The number s of linear forms is equal to

240

Chapter

5. Functions

That

Satisfy Linear

= $P

Differential

+ O(P-l)

Equations

.

If we substitute fi(o) in place of zi, these linear forms become numbers whose absolute values do not differ very much from ]P(fi(o), . . . , fm(a))I. It is easy to verify that the bound (46), when applied in this situation, gives the inequality in Theorem 5.29. The one drawback in the proof of Theorem 5.29 is that one does not obtain an effective expression for the constant c in terms of the degree d of the polynomial. For this reason the estimate in the theorem cannot be applied to polynomials with bounded coefficients and increasing degree d. The noneffectiveness of c is a consequence of the non-effectiveness of another constant - the o-, in the functional part of the Siegel-Shidlovskii method (see Lemma 5.2). The point is that, when proving Theorem 5.29, we apply Lemma 5.2 to the products of powers (47), and we want the lemma to give us a bound for ordR(z, fi(z), . . . , fm(z)), where R(xo,xI,. . . , x,) is a polynomial with complex coefficients. Clearly, in this situation the constant ~0 in Lemma 5.2 will depend on the system of equations that the functions (47) satisfy, and hence on d. If we impose some restrictions on the system of differential equations satisfied by the functions (47) (f or instance, the Siegel normality condition that we discussed in §2.3), then we can explicitly compute ~0, and hence the’ constant c in Theorem 5.29. However, this has not been done in the general, unrestricted case. In [1972] Nesterenko proved a generalization of Lemma 5.2 and used it to determine how c depends on d. The details appeared in [Nesterenko 19771, where it was proved that under the conditions of Theorem 5.29 one can take b = 4”h”(mh2

+ h + 1)

and

c = exp(- exp(rd2* In(d + 1))) .

Here it was assumed that each of the functions satisfies the modified definition of an E-function (see above). The constant r in the expression for c depends only on certain properties of the differential Galois group of the system (44) in Theorem 5.29. In many casesit can be explicitly determined - for example, if every nonzero solution of the corresponding homogeneous system m

YL = c

Qki(Z)Yi,

k= 1,2 ,...,

m,

i=l

consists of functions that are algebraically independent over C(z) (see [Brownawell 1985]), or if every solution of this system that has nonzero components is made up of functions that are algebraically independent over Cc(z) (see [Nesterenko 1988b]). Shidlovskii proved a homogeneous analogue of Theorem 5.29, in which he essentially finds bounds for the algebraic independence measure of the ratios , m. He also obtained bounds for polynomials in the fi(Q)/fl((y), i = %... values of E-functions that are connected by algebraic relations over (c(z). Detailed proofs of these and related results can be found in [Shidlovskii 19811.

$6. Bounds for Linear Forms that Depend on Each Coefficient

241

As examples of the consequences of these theorems, one obtains a bound for the transcendence measure of tana for nonzero cx E A, and also a bound for the algebraic independence measure of sin (~1, . . . , sin cr, for algebraic numberscri,...,o, that are linearly independent over Q. Shidlovskii also proved results when one has several sets of E-functions, each of which is a solution of a corresponding system of differential equations of the type (44). These and other results in this area are described in his book [Shidlovskii 1989a]. We also mention some quantitative results on the zeros of E-functions. In [1970b] Galochkin studied the arithmetic properties of the roots < of equations of the form P(z, fr(z), . . . , fm(z)) = 0, where P is an irreducible polynomial with algebraic coefficients and the fi(z) are algebraically independent E-functions that form a solution of the system (6). Shidlovskii’s fundamental theorem implies that c is transcendental. Galochkin obtained a bound for the transcendence measure of such numbers [ under the assumption that the height of the polynomial in the estimate is greater than some bound that depends (non-effectively) on its degree. This result was subsequently refined by VZininen [1977], Ivankov [1985], and Galochkin [1992].

$6. Bounds for Linear Forms that Depend on Each Coefficient In certain situations - for example, when estimating the deviation of a uniformly distributed sequence - one needs bounds on linear forms in several numbers that depend on all of the coefficients. From metric considerations (see Theorem 13 of [Sprindzhuk 19791) it follows that for any E > 0 and for almost all 3 = (&,... , S,) E ll? in the sense of Lebesgue measure, there exists a constant c = ~(6, E) > 0 such that for any nonzero vector h = (he,. . . , h,) E Zs+’ one has Iho + hl?91 + . . . + h,6,1 > c fr

ii-‘(ln(I

+ h))-“-”

,

(49)

i=l

where & = max(1, Ihi]) and h = maxrci<, ii. However, know a single s-tuple ?I for which this inequality holds.

at present we do not

.6.1. A Modification of Siegel’s Scheme. The following considerations can be used to prove this type of estimate. Suppose that we want to find a lower bound for ]l(~)] for a vector w = (~1,. . . , w,) with nonzero components, where the linear form 1(z) has the form I = Z(T) = hlccl + * * * + h,z,, Now suppose that for some positive

hi E Z,

h=maxhi.

X, p, Al, . . . , A, such that

i

242

Chapter 5. Functions That Satisfy Linear Differential

Ai& 5 Ai 5 ,u&, we have been able to construct

Equations

l
a set of linearly

Li = Li(5) = eCLijXj,

(50)

independent

linear forms

i=l,...,m,

j=l

with

oij E Z, so that the following

IL&z)]

conditions

< qXA fi

A;‘,

are fulfilled:

l
(51)

i=l

where cl = &

min lwil . . I Just as we did at the beginning of 56, we can choose m - 1 of the forms say, Ls,... , L, - in such a way that they together with 1 form a complete system of linearly independent forms. Let A be the determinant of the coefficient matrix of this set of forms (where the coefficients of 1 are in the first row of the matrix), and let Aij be the ij-cofactor of the matrix. We easily see that lAlj( 5 (m - l)!(Al * * * &)/Aj 7 (52) I&j1 Let k be a subscript

L (m - l)!(Al for which

i>2.

..-&)/(XAj),

(53)

A = Ak. If we replace a: by W in the identity

Axk = A,#i$

+ 2

AikLi@)

i=2

IAl 2 1 and (50)-(53),

and use the inequality

Zclm! 5 Iwk,I 5 (m - l)!(Ai

we find that

. . .A,)A-lI1(~)l

+ m!q

.

Hence, if we take into account that

A-’ fi

Ai 5 AJ’ fJ Ai 5 prn-‘h-’

i=l

with

r given by the condition

i=l

ii

i=l

that A, = h, we obtain

Il@)I 2 clA(A1 . ..A.,,)-’ For example, inequality

fi

2 clp l-“h(j&

. . . j&)-l

.

(54)

if m = s + 1, wm = 1, and wi = r9i for 1 5 i 5 s, we obtain the

$6. Bounds

for Linear

Forms that Depend

on Each Coefficient

243

When p = 1, this is precisely (49). In general, we see that the smaller the p we are able to use in (50), the better the estimate in (54). 6.2. Baker’s Theorem and Other Concrete Results. In [1965] Baker proved a lower bound of type (54) for the values of et at distinct rational points. Theorem 5.30. Let (~1,. . . , a, be distinct nontero rational numbers. Then for any nonzero integers hl, . . . , h, one has lhleal + . . . + hmeL1,12 clhl. . . h,l-’

h1-E(h) ,

where h = maxi
and ord,=eR(z)

2 ~1 + . . . + T, + m - 1 -

where

[& 1 -

R(z) = PI (z)eal* + . . . + P,(z)e”-*

.

Here and later, the ck are positive constants that depend only on m, cyi, . . . . . ..am. This construction and the technique for subsequently choosing linearly independent numerical forms are similar to the constructions in Siegel’s method. By differentiating R(z) a certain number of times and substituting z = 1, we can obtain linearly independent forms Li(:), 1 5 i 5 m, that satisfy (50) and (51) with Aj = rj!cs& for certain rj E N, X = c;&, and p = r-X. The bound in Theorem 5.30 then follows from (54). In [1967a] Fel’dman proved a result similar to Theorem 5.30 for 1 and the values at (Y E Q, cy # 0, of the functions cpxl (z), . . . , cp~,,,(z), where

cpx(z> = 1+ n=l 2

(A + 1) .“.: (A + T8) ’

with Xi,.. . , A, E Q distinct modulo 1. The coefficients of Pi(z) in the corresponding functional linear form were given explicitly, and this made it possible to improve the final inequality, replacing E(h) by a somewhat smaller function E(h) = v(ln In h)-’ . In [1972] Fel’dman used explicit constructions of the polynomials Pi(z) to prove a theorem similar to Theorem 5.30 for a linear form in numbers that themselves are linear combinations of logarithms of algebraic

244

Chapter

5. Functions

That

Satisfy Linear

Differential

Equations

numbers. Here c(h) > 0 is arbitrarily small but fixed, i.e., it is essentially independent of h. This weakening of the result is related to the fact that the functions in the proof are not Efunctions, but rather belong to the broader class of G-functions (see 57). This result was generalized by Sorokin [1985], who proved a similar bound for the values of a set of functions

where

A(),... , &n E Q,

x0, . . . , Ln > 0,

Xi f Xj (mod 1) for 1 5 i < j 5 m .

If Xc = 1, these functions are the same as the ones in [Fel’dman 19721. Sorokin’s proof is based on an effective construction of Pad6 approximations of the second kind for these functions (see Chapter 2). 6.3. Results of a General Nature. In [1974] Galochkin announced a general analogue of Theorem 5.30 for a set of G-functions each of which is a solution of a first order linear differential equation with coefficients in C(z). He also assumed that the set of functions has the so-called G-property (see §7), and the values are taken at a point of the form l/q, where q is an integer in a quadratic imaginary field. The details of the proof were published in [Galochkin 19751. Bounds for linear forms in ln(1 - Qi/q) and for linear forms in (1 + al/q)“l a.. (1 + am/q)“- for distinct m-tuples ~1, . . . , vrn E N were obtained as corollaries. In [1976a] Vaananen, using the same ideas as Baker, generalized Theorem 5.30 to the (m + 1)-tuple of functions 1, cp~(a;.z), 1 5 i 5 m, where X E Q \ Z are distinct nonzero rational numbers. His function e(h) has andcrl,...,o, .... .. the sameform as in Theorem 5.30. In [1976b] Vaananen proved a result similar to the theorem of Galochkin mentioned above for the values of any set of Efunctions that are linearly independent over C(z) and have Taylor coefficients in a quadratic imaginary field K. Here each E-function must be a solution of a first order linear differential equation with coefficients in C(z). The following more general theorem was proved by Makarov [1978]. Theorem

5.31.

Suppose that the E-functions fll(z>,...,flnl(Z),...,fml(Z),...,fmn,(Z)

are, together with 1, linearly independent over C(z); and suppose that they form a solution of the system of differential equations Y:j = Qijo(z) + 2 Qijs(Z)Yis, &?=I

i=l

,...,m,

j=l

,.-.,ni,

with Qijs(z) E C(z). Further suppose that K is a quadratic imaginary field, and cr E R is not equal to 0 or a pole of any of the Qijs(z). Then there exists

$6. Bounds for Linear Forms that Depend on Each Coefficient a constant c > 0 depending only on cr and the functions fij(z), j = l,..., ni, such that for any hij E Z not all zero, one has

245

i = 1,. . . , m,

-1

h-“(h) where hi = ma.xl<j
Ihijl,

h = maxi
Ihil, and e(h)

, is

the same as in

We note that both in this theorem and in Vaananen’s result mentioned above, one uses the version of the definition of an E-function with a stricter condition on the growth of the conjugates of the Taylor coefficients and their denominators (see the remark before Theorem 5.27). The proof of Theorem 5.31 uses ideas from the Siegel-Shidlovskii method and also [Baker 19651. Unfortunately, the bound in the theorem does not depend on each of the coeflicients of the linear form, but only on the maximum modulus of the coefficients of the groups of variables corresponding to the systems of differential equations. In [1984] Makarov derived some consequences of this theorem relating to bounds on polynomials. In [1984a] Chudnovsky published the following result: Let

be E-functions with of a linear differential and any nonzero r 0 with the following max(]Hi], . . . , IHmI)

rational Taylor coeficients, each of which is a solution equation with coeficients in C(z). Then for any E > 0 E Q there exists a constant ~0 = co(e, r, fl, . . . , fm) > property. For any nonzero HI,. . . , H,,, E Z with H = > ~0 one has either Hlfl(r)

+ . . . + H, fm(r)

= 0 ,

or else IHlfl(r)

+..s+

Hmfm(r)l

> IHI...H,I-~H~-~.

It should be noted that what Chudnovsky actually proves is not this result, but rather a weaker assertion in which several additional conditions are imposed on the functions fl(z), . . . , fm(z) in the course of the proof. The proof is carried out under the assumption that the so-called conjugate differential equations to the ones in the theorem satisfy the Siegel normality condition. It is further assumed that r is not a singular point of any of the systems of differential equations. The result has not yet been proved in the version given above. We also note that the conditions imposed on the fi(z) make it impossible to include the derivatives of any of these functions in the set of functions. In the special case fl(z) = 1, fa(z) = KA(z) (see §l), Titenko [1987] was able to refine Chudnovsky’s argument and prove the following result.

246

Chapter 5. Functions That Satisfy Linear Differential

Equations

Theorem 5.32. Let X E Q, X # O,-l,-2,.. ., and X # (2k + 1)/2 for any k E Z; and let a be a nonzero rational number. Then for any p, q E Z, IqI > go > 0, one has lPKx(~) where go and y are positive A substantial

+ 91 >

Iql- 1 - -#nln

constants

generalization

1q1)-“3

7

that depend only on X and cr.

of this theorem was obtained

$7. G-Functions

in [Zudilin

19951.

and Their Values

7.1. G-F’unctions. The Siegel-Shidlovskii method can be applied to the values of the generalized hypergeometric functions al,...,al

O”

IF, h,.

“,

bm

”

=nTo

(al>,..-(ad,

(bl)n...(bm)nn!Zn

in the case 1 = m + 1, i.e., when the power series has finite radius of convergence. As before, we assume that ai, bj E Q. In [1929] Siegel defined the class of G-functions, which includes the generalized hypergeometric functions with 1 = m + 1, and discussed the possibility of studying the values of G-functions at algebraic points. Here is the exact definition. Definition

5.4. An analytic

function

f(z) = 5 c,zn n=O is said to be a G-function 1) 2)

if the following

are fulfilled:

all of the coefficients cn lie in an algebraic number field lK of finite degree over Q; for some constant C > 1 and any n 2 1 we have l-z-l

3)

conditions

SC”,

where m denotes the maximum modulus of the conjugates of cr; and there exists a sequence of natural numbers ql,q2, . . ., where qn 5 C”, such that for all n for O<j
The G-functions form a ring that is closed under differentiation and under integration from 0 to z within the disc of convergence. Besides the generalized

$7. G-Functions

and Their

Values

247

hypergeometric functions with rational parameters and finite radius of convergence, this class also contains algebraic functions whose Taylor series have algebraic coefficients (as one seesfrom the Eisenstein criterion). As in the case of Efunctions, in order to apply the Siegel-Shidlovskii method we must have a linear differential equation with rational coefficients that is satisfied by the function under consideration. The study of the values of G-functions is much more complicated than the situation with Efunctions. One cannot expect to obtain general transcendence results for the values of such functions of the type that one can prove for Efunctions. The algebraic functions mentioned above as examples of Gfunctions (such as G) take algebraic values at algebraic points. In fact, the irrational function G takes rational values at an infinite sequence of rational points, namely, z = 1 - r2 for r E Q. In the case of G-functions, there are technical difficulties in the proofs resulting from the much slower convergence of the series compared to Efunctions. It turns out that the functional linear forms with high order zero at z = 0 that are constructed in the Siegel-Shidlovskii method are not very small when one substitutes a concrete value z = a. The required small bound on the size of the numerical linear forms at (Y can be proved only for cr near eero. This explains, for example, why the Siegel-Shidlovskii method cannot be used to prove irrationality of the values of the Riemann zeta-function at odd integers. Namely, C(2k

+

1)

=

J52k+l(l)

,

where the polylogarithm

belongs to the class of G-functions. It should also be noted that Fermat’s Last Theorem is an assertion about irrationality of the values of the G-function

at all rational points z, 0 < z < 1. Siegel himself announced several partial results, for which he did not supply complete proofs, but only indicated that they can be proved using his method. For example, he stated that

is irrational for x = p/q E Q satisfying the inequalities 0 < ;
10-m,

248

Chapter 5. Functions That Satisfy Linear Differential

Equations

where cl is a positive constant. Here is another result that Siegel announced. Let y be an algebraic function of x that is regular at x = 0 and whose equation has algebraic coefficients. Further suppose that the abelian integral Jo” y dx is not an algebraic function, and < is a nonzero algebraic number of degree k that is a root of a polynomial whose rational integer coefficients are no greater than M in absolute value. If for some E > 0 one has [[I < c2 exp

(

-(lnM)li2+&

1

,

where cp is a positive constant depending only on E and lc, then the number s,’ y dx does not satisfy any algebraic equation of degree k with rational coefficients. In order to find a general scheme for assertions of this type, let us consider the preceding result of Siegel. Let m denote the degree over Cc(x) of the algebraic function y, and set O<j<m;

fj(X> = Y(X)?

fm(x)

= I”

ydx .

As mentioned before, these functions are G-functions; and they are linearly independent over e(z). In addition, they form a solution of a certain system of linear differential equations with coefficients in Cc(x). The claim that the numbers f?(t), 0 5 j 5 m, are linearly independent over a certain field means, in particular, that s,’ y dx does not belong to that field. This is what Siegel stated. It is also easy to see that the first result of Siegel mentioned above follows from the linear independence over Q of the values of the G-functions I,

Z~l ( 112$‘2;z)

,

and

zFi ( 1’2f’2;z)

at a point x = p/q that satisfies the given conditions. The first general theorems on the values of G-functions were proved in [Nurmagomedov 19711. Among other results, he showed the following. Suppose that fi(z),... , fm(z) are G-functions that are linearly independent over c(z) and satisfy the system of linear differential equations Y;

= 5

Qki(z)~i,

k=

1,2 ,...,

m,

(56)

i=l

where Qki E Cc(z). Let (Y E A and b, H E IV, where b is greater than an explicit constant bo that depends only on H, cr, and the functions fi(z). Further suppose that a/b is not equal to a singular point of the system of differential equations. Then the numbers fi(cr/b), . . . , fm(a/b) are not connected to one another by a linear equation with rational integer coefficients of absolute value not exceeding H. Under similar conditions he proved a lower bound for the

$7. G-Functions

and Their

Values

249

absolute value of a linear form in these numbers; and he obtained analogous results for polynomials. In [Chirskii and Nurmagomedov 1973a, 1973b] and [Chirskii 1973b] the authors applied these results of Nurmagomedov to functions zFi (y ;z) with rational parameters, their derivatives, and elliptic integrals in Legendre form. A serious drawback of these results is that the lower bound be for the denominators of o/b depends on the parameter H, i.e., on the size of the coefficients of the linear form. In other words, for a fixed point a/b one is able to prove only that there are no relations with coefficients of bounded size among the numbers fi(cr/b), . . . , fm(o/b). This does not, however, imply that the numbers are linearly independent. Moreover, in [1978a, 197813, 1978c] Chirskii showed that equally strong results can be proved for a broader class of functions that includes both Efunctions and G-functions. In particular, such results hold for hypergeometric functions with irrational algebraic parameters, and also for a set of functions consisting of eaiz andln(l-pja)with~li,pjEA,lLiIm,lIjLn. In order to understand the reason for this deficiency in the results, we briefly recall the argument. First, in the Siegel-Shidlovskii method one constructs a functional linear form R(z)

= fi(Z)fl(Z)

+***+

P*f*(z)

,

where Pi(z) E Z,[z], that has a zero at z = 0 of sufficiently high order compared to the degrees of the Pi(z) and the size of their coefficients. One then constructs a sequence of functional linear forms Rk(Z)

= Pkl(Z)fl(Z)

+-..+Pkm(Z)fm(Z),

k>l,

with Pki(z) E C(z), that have high order zeros at z = 0. The operation of producing new linear forms can be conveniently interpreted as an action of the differential operator

on the linear form R = Pl(z)zl

+ -.. + Pm(z)z,

.

Then RI(Z) = R,

%+1(Z) = (t(z))“D”R(Z),

k= 1,2,...

.

Here t(z) is a common denominator for the coefficients of the system of differential equations (56). We have somewhat modified the method of constructing the forms Rk that was used in our study of E-functions. Next, one can prove that there are m linearly independent numerical linear forms among the forms

250

Chapter 5. Functions That Satisfy Linear Differential hl

(ah

+

. . . +

&I

(Q>Gn

9

Equations

l
These linearly independent forms are the ones that are used in the rest of the proof. The same type of argument works for G-functions. However, in that case, because of the slow convergence of the power series, one must be concerned with the range in which the index of the functional linear forms can vary. then the coefficients of the Ic-th derivative If P(z) E e[ z ] is a polynomial, Pck) (z) have numerical factors that are > k!. A similar situation occurs when one computes the forms &+i(Z) = ~(z)~@R(Z). As Ic takes values of order in, the k! factors in the bounds for the coefficients of the polynomials Phi(z) have a magnitude comparable to that of the linear forms being constructed. In the case of E-functions these terms are small enough to be neglected; but when we study G-functions it is precisely these terms that force us to impose restrictions on the growth of the coefficients of the linear forms. 7.2. Canceling Factorials. When he announced his results on G-functions in 1929, Siegel mentioned in passing that one can take out a common factor in the coefficients of the forms. Most likely, what he had in mind was that the coefficients of the polynomials &(z), 1 5 i 5 m, must have a common factor roughly equal to k!. If we remove this common factor, then as before we obtain linear forms with integer coefficients, but the bounds on the coefficients are much better as a function of Ic. There were difficulties in implementing this idea. In [1974] Galochkin axiomatized the relevant property, and introduced the subclass of G-functions for which the property holds. Definition

5.5. A set of G-functions

that forms a solution of the system of differential equations (56) is said to have the G-property if there exists a sequence of natural numbers a,, n > 1, such that for all n > 1 and for any m-tuple of polynomials PI(Z), . . . , P,(z) E &[z] the coefficients of the linear forms ~(t(z))“D”R(Z),

k= 1,2 ,..., n,

are polynomials in ZK[Z], and one has Ian] 5 cn for some constant c 2 1. In [1974] Galochkin proved that products of powers of ln(1 + Q~z), cri E A, have the G-property, as do products of powers of (1 + Q~z)“’ for algebraic CQ and rational Vi. In particular, he proved the following theorem. Theorem 5.33. Suppose that H E IR, q, d E N, and the functions f~ (z), . . . . . . , fm(z) have the G-property and are not related to one another by any nontrivial algebraic equation of degree at most d with coeficients in Cc(z). Further suppose that P E Z[xl, . . . ,x,1, P f 0, is a polynomial of degree at

97. G-J?unctionsand Their Values

251

most d whose coeficients have absolute value at most H. Then there exist positive constants A = A(K, fi, d), X = X(K, fi, d), and /I = p(lK, fi, d, q) such that for q > A one has > q-XH-p

.

The constants A and p, which depend upon the parameters in a rather complicated way, are given explicitly in [Galochkin 19741. In particular, Galochkin proves results for ln(1 + al/q), . . . , ln(1 + a,/q) for pairwise distinct oi,...om E A. For example, if q > e7g5, then ln(l-~)ln(l+~) is irrational. If d = 1, then Theorem 5.33 gives us linear independence of 1, fiWq>, . . . , fmWq) over Q. In [1975] Galochkin proved a result analogous to Theorem 5.30 - i.e., a bound on linear forms that takes into account the rates of growth of all of the coefficients - for the values of any set of G-functions that have Taylor coefficients in a quadratic imaginary field K, are (along with 1) linearly independent over e(z), and satisfy the G-property. Here each of the G-functions must be a solution of a first order linear differential equation with coefficients in Cc(z) (see IS). Galochkin applied this result to the values of ln(1 - criz) and products of powers of (1 + (Y~z)~‘. We note that in [1972] Fel’dman had proved such a result for the functions

which also satisfy first order differential equations. In [1981] Matala-aho and V&%-&en proved that the function

F(z)= r’2f’2;z) 2F1

and its derivative have the G-property. Using this, they obtained lower bounds for polynomials in F(cr/q) and F’(a/q) with integer coefficients, provided that cr and q satisfy certain conditions. In particular, they showed that at least one of the numbers FCp/q) or F’(p/q) satisfies the assertion announced by Siegel with a constant C in place of 10. In [1983] Beukers, Matala-aho, and VBananen proved that the functions 2F1

and

2F,’

have the G-property for any (Y,p, y E Q with y # 0, -1, -2,. . .. In the same paper they proved a theorem similar to Theorem 5.31 for the values of

252

Chapter 5. Functions That Satisfy Linear Differential Equations

where cri, /3i, yi E Q satisfy certain natural conditions, at a point p/q E Q, where q > qo = q,-,(p). This result implies, in particular, that these values are linearly independent over Q. 7.3. Arithmetic Type. In [1981] Bombieri proposed another condition that can take the place of the G-property and similarly make it possible to use the Siegel-Shidlovskii method for G-functions. Let lK be an algebraic number field of finite degree over Q such that the coefficients Qij (z) in the system (56) lie in K(z). Bombieri’s approach is based on studying the properties of the solutions of (56) over all of the completions of the field K. For each absolute value 1 IV on lK we let K, denote the completion of lK with respect to v, and we let R, denote the completion of the algebraic closure of KU. We also use the notation 1 Iv for the extension of the absolute value to 0,. If we fix a point < E fl,,, we can consider the solutions of the system of differential equations (56) in the ring of formal power series in z - < with coefficients in a,. If Iz - 0 by setting log+(z) = In max(1, x), and Vo is the set consisting of all non-archimedean valuations on K. If the inequality (57) holds for the system (56), then the system (more precisely, the differential operator corresponding to the system) is called a Fuchsian operator of atithmetic type. In Andre’s book [1989] the term G-operator is also used. The condition (57), which replaced the condition on canceling factorials, enabled Bombieri to apply the Siegel-Shidlovskii method and prove several general theorems on the values of G-functions. Here we shall only give two consequencesof these theorems that concern algebraic functions and polylogarithms. Theorem 5.34. Suppose that K is an algebraic number field, d = [IK : Q], and u(z) is an algebraic function of one variable that is regular in a neighborhood of 0 and satisfies the relation P(z,u) = 0, where P is a polynomial with rational coeficients. Further suppose that s: u(t) dt is not an algebraic function. Then there exist two constants cl and ~2, depending only on u(z), with the following property. If < E K, c # 0, h(J) > cl, and

$7. G-Functions

and Their

253

Values

ICI < exp(--c2dJin h(J) lnln h(C)) , where ) 1 is the usual absolute value in C, then the complex numbers,’ u(t) dt does not lie in K. (Here and later h(J) denotes the so-called Yogarithmic height” of the algebraic number [.) This theorem is a refinement of the result on the values of abelisn integrals that was announced by Siegel. Condition (57) can easily be verified using the Eisenstein criterion. Theorem 5.35. Suppose that K is an algebraic number field, v is a valuation of K, and [ E II6 satisfies ]
L:(z)= 2 f,

k= 1,2 )...) m,

n=l

be the polylogarithms. Then there exist two constants cs and ~4, depending only on m, with the following property. For any 6 E N, if the conditions h(t) > exp (c3(m)62m+1 ln(6 + 1)) ,

151v< exp( -~4(m)61~(2m)(lnh(~))1-1~(2m)(lnlnh(J))1~(2m)

> are fulfilled, then the numbers Ll(J), . . . , L,(J), regarded as elements of K,,, do not satisfy any polynomial equation of degree6 with coeficients in K. 7.4. Global Relations. Bombieri’s approach relates the local and global properties of the solutions of differential equations. In particular, in [1981] he investigated the question of which G-functions have “global relations.” Definition 5.6. Suppose that fi(z), 1 5 i 5 m, are formal power serieswith coefficients in the field IK, c E K, and Q E lK[xi, . . . ,x,1. A relation

Q(fi(C), . . . 7fm(C)) = 0 is said to be global if it holds in K,, for every place v of lK for which I<],, is less than the v-adic radius of convergence of all of the series fi(z). For example, consider f @) = (1 - 4-w

= 2 (2n)!zn n=O4%!)2

)

where K = Q and < E Q. It is easy to see that the series f(t) provided that

IEI -c 1 I% -c 1

for for

converges in U&

p = c0, i.e., U& = IR ; P>2;

and

1 1512< -4 ’

254

Chapter 5. Functions That Satisfy Linear Differential

Equations

For instance, for t = 8/9 the series f(t) converges in lR and Q2. Note that f(8/9) converges to +3 in lR and to -3 in Qz. The relation f(c) - 3 = 0 is not global, since it does not hold in Q2. If we take c = -3, then f(t) converges only in Qs, where it converges to -l/2; hence f(c) + l/2 = 0 is a global relation for t = -3. In [1981] Bombieri proves the following general result about global relations among the values of G-functions. Suppose that our system of differential equations is Fuchtype, i.e., the property (57) holds; and suppose that (fl(ZL...> fm(z)> is a solution consisting of G-functions that are linearly independent over A(z). Then the points E E A where there are global linear dependence relations among the numbers fi(<), . . . , fm(<) satisfy the inequality h(t) < ~5 , Theorem

5.36.

sian of arithmetic

where c5 is a constant depending only on the system (56) and the functions

fib). This theorem trivially

implies

Corollary 5.2. There are only finitely many algebraic points 5 of bounded degree where there are global linear dependence relations among the numbers

fl(C** *,fm(5). We note that the constant cg in Theorem 5.36 can be effectively determined, and thus one can find all such points [ whose degree is lessthan some bound.’ In [Bombieri 19811similar results are proven for global polynomial relations. We now give an example (taken from [Bombieri 19811) of how Theorem 5.36 can be used to solve Diophantine equations. Consider the equation x4 - 4xsy2 + 4y4 - 2x2 - 4y2 + 5 = 0,

(59)

which determines an algebraic curve of genus 1. Already in [1929/1930] Siegel proved that any equation of the form f (x, y) = 0, where f is a polynomial with rational coefficients and the corresponding curve has genus 2 1, has only finitely many integer solutions. In [1970] Baker and Coates found explicit bounds on the magnitude of solutions. We show how this can be done using Theorem 5.36 in the specific case of the equation (59). This equation can be rewritten in the form (1 - x2 - 2y2)2 - 2(2sy)2 = -4 ) or, if we set q = x + yfi in the form

and [ = 1 - q2 and assumethat x and y are integers, Norm(<) = -4 ,

where Norm(<) is the norm from Q(a) to Q. Thus, the problem of finding all integer solutions of (59) reduces to finding all t, 77E Z[Jz] such that

$7. G-Functions

and Their

Values

J=l-72,

Norm(t) = -4 .

255

If a pair (J,n) satisfies these conditions, then so do the pairs (<, -n), (<‘,n’), and (<‘, -n’), where <’ and 7’ are the conjugates of < and n in Q(d). Hence, without loss of generality we may assume that 0 < < < 1 and n > 0. Then [’ = (-4)/t < -4. Th is means that the series (58) diverges at c’ and converges at’[, where its value is f(c) = n. If we set fi(z) = 1 and fs(z) = f( z ) , we obtain a pair of G-functions that form a solution of the system of differential equations Y; = 0,

1 Y4 = q1 _ z)Y2 7

for which one easily verifies that (57) holds. According to Corollary 5.2, there are only finitely many 5 E R = Q(a) such that there is a global linear relation between fi (I) = 1 and f2 (0. We now show that any pair (J,n) coming from a solution to (59) as explained above, gives rise to a global linear relation

It suffices to show that the series f(c) diverges in all non-archimedean completions K, of K. Since < and <’ are algebraic integers and Norm(<) = -4, it follows that ]& = 1 for any non-archimedean valuation v with v]p, p > 2, and hence f(t) diverges in I&,. There is only one place of lK above p = 2; we let ) 12 denote the corresponding absolute value. Then It]2 = ]<‘]2, and so ICI2 = l/2 ( since t<’ = -4), i.e., f(c) also diverges in the completion IQ, corresponding to the absolute value ( 12.From Theorem 5.36 it now follows that there are only finitely many points < with the above properties, i.e., only finitely many solutions of the Diophantine equation (59). In [1990a, 1990b, 19921 Chirskii studied global relations among values of functions that are not G-functions. He proved that there are no global relations in the class of functions he considered. This led to effective bounds for prime numbers p with the property that a given algebraic relation is violated in K,, for vlp. 7.5. Chudnovsky’s Results. In [1984b] Chudnovsky proposed another approach to proving results about G-functions. In some sense Chudnovsky’s method is dual to that of Siegel-Shidlovskii. It avoids the necessity of canceling factorials or working only with systems of differential equations that are of arithmetic type. Moreover, Chudnovsky was able to prove that the factorial canceling property always holds (see Theorem 5.38 below). Here is the fundamental result of [Chudnovsky 1984b]. Theorem 5.37. Suppose that the G-functions f~ (z), . . . , fm(z) are linearly independent (along with 1) over Q(z) and form a solution of a system of linear differential equations (56) with coeficients in Q(z). Then for any E > 0 and for any nonzero r = a/b such that a E Z and b E N satisfy the inequality

256

Chapter 5. Functions That Satisfy Linear Differential (f

2

Cllap+l)(n+4

Equations

,

the numbers 1, fi (r), . . . , f,,,(r) are linearly independent over Q. Moreover, any ho,. . . , h, E Z with H = max(lhol,. . . , Ihml) > 122one has Iho + hlfi(r)

+ ... + h,f,(r)I

for

> HmrnmE .

Heretheconstantscl=c~(f~,...,fm,&)>Oandc2=c2(f~,...,fm,&,r)>O can be effectively determined. Instead of functional linear forms in fr (z), . . . , fm(z) with a high order of zero at z = 0 (i.e., Pad6 approximations of the first kind), the proof of this theorem uses so-called Pad6 approximations of the second kind (see Chapter 2). We briefly describe the basic ideas in the proof. Definition 5.7. Suppose that the functions gr (z), . . . , gm(z) are analytic at z = 0. Let M and D be nonnegative integers. By a system of Pad6 approximations of the second kind of type (D, M) for the functions gi(z) we mean a set of polynomials Q(Z) , Pr (z) , . . . , P,(z) E C[z] of degree at most D that satisfy the conditions

ord,,o(Q(z)gi(z)

- Pi(z)) 2 D + M + 1,

i=l,...,m.

Using the Dirichlet pigeonhole principle, one easily verifies that if D 2 mM, then Pad6 approximations of the second kind always exist. For our later purposes it is very useful to note that the polynomials Pi(z) are uniquely determined by the polynomial Q(z) in any system of Pad6 approximations. Thus, when working with different Pad6 approximations in the proof of Theorem 5.37, we need only control the size H(Q) of the coefficients of Q(z) E Uj[z] and the size of a common denominator of those coefficients. Then the bounds on the polynomials Pi(z) will be determined by the corresponding bounds on Q(z). Now suppose that the conditions of Theorem 5.37 are fulfilled, D is a sufficiently large integer, and S E lR satisfies the inequalities 0 < 6 < l/m. We and use the Dirichlet pigeonhole principle to show that set M= [(A-6)D] there exist Pad6 approximations (Q(z), PI(Z), . . . , P,(z)) of type (D, M) for the functions fi(z), . . . , fm(z), where Q(z) E Z[z] and

withcs=cs(fi ,..., fm,m,6)>0. We next use a procedure for elimination of variables that is reminiscent of Hermite’s proof of transcendence of e; here the numbers Pi(r)/Q(r) play the role of simultaneous rational approximations to fi(r). As in Hermite’s method, one needsto construct a complete system of independent approximations. This is accomplished by differentiating the forms Ri = Ri(z) = Q(z)fi(z) - Pi(z) that we already have, and eliminating “extra” functions from R:(z). We set

$7. G-Functions j$‘)

z

= R.

and Their Values

257

l
27

and lli<m, Using induction

k>O.

on k, one easily verifies that Z”)(z) z

= &‘k’(Z)fi(z)

- F!“‘(z) z

)

where Q(k), 5”’2

E Q[z] >

i=

l,...,m,

and

@“+‘)(z)= -&&‘k’(z)= ($) k+lQ(z) ~‘k’l)(~) 2

=

$~~k’(~)

-

Qij(z)~~k’(Z)

~

,

.

j=l

Let t(z) E Z[z] b e a common denominator for all of the coefficients in the system (56), and let d = maxdeg,(t(z),t(z)Qij(z)). For k 2 6 and 1 5 i 5 m we set Q(k)(~)

PJk’(z) *

= t(~)“A@~)(z), k!

Rck)(z)

= t(z)k;i?k)(z)

= t(&3k~(z) k!”

= Q’“‘(z)fi(z)

’

- P,!k”(z)

.

One can verify that for any k 2 0: i)

ii)

the set of polynomials (Q(“) (z), I’{“’ (z) , . . . , Pg’ (z)) is a Pade approximation of the second kind of type (D + kd, A4 - k(d+ 1)) for the functions f&l; Q(“)(z) E Z[z] and P,(‘)(z) E Q[ z ] , w h ere the least common denominator of all of the coefficients of the polynomials P:“‘(z) is at most c:+~, c4 = c4(f1,...,fm,77h6)

iii)

>O;

the following

inequalities H(Q’k’)

IR!k’(z)j 2 For D sufficiently

are satisfied: 5 c;+~ 7

= IQ’k’(~)fi(z)

H(P!k’)z

- P!“)(z)l t

large the determinant

II

Qck’(z),

Pik’(z)

<- ,;+I,

,

5 c~+~IzI~+~+~-~

A(Z) of the matrix

7* . * 7

p’)(z)~~O
--

.

258

Chapter

5. Functions

That

Satisfy Linear

Differential

Equations

is nonzero. This can be shown in much the same way as in the proof of Lemma 5.2. The nonvanishing of this determinant implies that there exist integers Icj, 0 5 k. 5 . . . 5 k, 5 m8D + cg, such that det Q@j) (r), P..“j) (r) >***, II Because of iii) above, all of the values ]R$$‘(r)) are small. The rest of the argument is just like Hermite’s classical proof of the transcendence of e. The ideas developed by Chudnovsky to prove Theorem 5.37 enabled him to show that the G-property holds in general for any set of G-functions that form a solution of a linear system of differential equations with rational coefficients (see [Chudnovsky 1985c]). Theorem 5.38. Suppose that the G-functions fi (z), . . . , fm(z), m 2 2, are linearly independent over C(Z) and form a solution of the system of differential equations (56). Then these functions have the G-property. The topic of G-functions was further developed in [Andre 19891. In this book Andre used Chudnovsky’s results to put Bombieri’s results in a more precise form, and also to discussthe connections between the arithmetic properties of the values of G-functions and various questions in Diophantine geometry. One of the results in the book is that the factorial canceling properties of Galochkin (see $7.2) and Bombieri’s arithmeticity of the differential operator of the system of equations (see 57.3) are equivalent.

Chapter 6. Values of Functions That Have an Addition Law

259

Chapter 6 Algebraic Independence of the Values of Analytic Functions That Have an Addition Law In this chapter we are interested in algebraic independence of the values of the exponential function and the Weierstrass pfunction. In the case of the exponential function with algebraic values of the argument, the situation is completely described by the Lindemann-Weierstrass theorem. However, transcendental values of the argument bring in a whole new series of difficulties. For example, Hilbert’s seventh problem asks about transcendence of the value of ez at the transcendental point z = /3 In a, where CY,/3 E A, (Y # 0, 1, and /3 is irrational. It seems that Gel’fond was the first to study algebraic independence of the values of the exponential function at points that are not necessarily algebraic. In this connection he made the following two conjectures. 1. Let a be an algebraic number not equal to 0 or 1, and let ,&, . . . , & be algebraic numbers such that 1, pi, . . . , /3,,, are linearly independent over Q. .Then the numbers are algebraically independent over Q. 2. Suppose that (~1,. . . , (Y, are nonzero algebraic numbers, and In al,. . . are branches of their logarithms that are linearly independent over “‘7 lna, Q. Thenlnai,... , In (Y, are algebraically independent over Q. Gel’fond never actually wrote down these conjectures; however, he frequently mentioned them in his talks at the number theory seminars at Moscow State University, and he undoubtedly tried to prove them. The first conjecture, which for m = 1 coincides with Hilbert’s seventh problem, is a natural analogue of the Lindemann-Weierstrass theorem. It still has not been proved for any m > 2. It is easy to verify that it is equivalent to the following conjecture, which at first glance seemsless general. 3. Let cx be an algebraic number not equal to 0 or 1, and let 0 be an algebraic number of degree d 2 2. Then the numbers

are algebraically independent over Q. It is this conjecture that was investigated by Gel’fond. In [1949b] he proved it in the case d = 3 (seeTheorem 6.3 below), and in [1949d] he obtained several more general results. The ideas in [Gel’fond 194913,1949d] turned out to be very fruitful, and the proofs of all of the results in this chapter are based on them. Concerning the second of Gel’fond’s conjectures, we note that all that is currently known is that 1, In (~1,. . . , In elm are linearly independent over the

260

Chapter

6. Values of Functions

That

Have an Addition

Law

field A of algebraic numbers. This was proved by Baker in 1967 in connection with bounds for linear forms in the logarithms of algebraic numbers (see Chapter 4). There is another conjecture on the values of eL at transcendental points that generalizes both the Lindemann-Weierstrass theorem and Gel’fond’s conjectures. It was first stated in Lang’s book [1966], although specialists knew of it before then. Schanuel’s Conjecture. Suppose that al,. . . , ap E Cc are linearly independent over Q. Then there are at least p of the 2p numbers al ,...,

ap,e a1,...,

e%

that are algebraically independent over Q. In other words, the transcendence degree over Q of

Q(al, . . . ,ap, eal,. . . ,e+)

(1)

is at least p. When all of the ai are algebraic, Schanuel’s Conjecture reduces to the Lindemann-Weierstrass theorem. If we take p = 2, al = 1, and a2 = 27~i, then the field in (1) is an algebraic extension of Q(e, n), and so the conjecture says that e and x are algebraically independent over Q - another known problem of transcendental number theory. This conjecture implies results that are even more general than Gel’fond’s third conjecture. Suppose we set p = d andai=@-llncr,i=l,. . . , d. Then the field in (1) is an algebraic extension of Q(lna,oB,c# ,..., crfld-l). Thus, Schanuel’s Conjecture implies that the numbers are algebraically independent over Q. The second conjecture of Gel’fond is also a consequence of Schanuel’s Conjecture. We now give some notation that will be used in this chapter. As before, if P E C[51,22,..., zm], then deg P is the total degree of P in ~1,. . . , 2,; H(P) is the maximum modulus of the coefficients of P (we call this the height of the polynomial P); and t(P) = degP +lnH(P). If K = Q(&,...,&) C Cc and if WI,..., wq is a maximal set of elements of K that are algebraically independent over Q, then we call {WI,. . . ,wp} a transcendence basis of K, and we call q the transcendence degreeof K. The transcendence degree does not depend on the choice of transcendence basis; it is denoted tr deg K.

$1. Gel’fond’s

Method

and Results

Because of the importance that Gel’fond’s results have had in the development of the theory, we shall describe them in some detail.

$1. Gel’fond’s

Method

and Results

261

1.1. Gel’fond’s Theorems. Schanuel’s Conjecture has been proven only in the case p = 1. In that case it is equivalent to Lindemann’s theorem on transcendence of e* for nonzero z E A. Because of the difficulties in the general case, it is useful to introduce simpler versions of the problem. Suppose that, along with al, . . . , up, we consider another set br, . . . , b, of complex numbers that are linearly independent over Q. Following Gel’fond, one customarily tries to prove bounds on the transcendence degree for three types of fields: Ll

= Q (ealbl,.

Lz =Q(al,..., L3 =Q(al,...,

. . , eapbq)

ap,ealbl ,..., aP,bl ,...,

,

eapbq) ,

bp,ealbl ,...,

eapbq) ,

generated over Q by pq, pq + p, and pq + p + Q elements, respectively. These fields arise naturally if we replace the functions z and e”, whose values at oi occur in Schanuel’s Conjecture, by the larger set of functions z, eblz, . . . , ebqz for some Q > 1. Note that Schanuel’s Conjecture is the case L2 with q = 1, bl = 1. In [1949d] Gel’fond proved the following results, in which the numbers oi and bj have to fulfill more stringent conditions than above. Theorem 6.1. Suppose that p = q = 3, the numbers al = l,as,as are linearly independent over Q, and the numbers bl, bp,bs satisfy the following condition: there exists a constant y > 0 such that for any xi, x2, x3 E Z with X = max(]xr 1,1x21,1x3I) suficiently large one has --yx

lxlbl + x2b2 + x3b31 > e

In x

(2)

Then the field L2 has transcendence degree at least 2. Theorem 6.2. Suppose that p = 3, q = 2, the numbers al = 1, a2, as are linearly independent over Q, and the numbers bl # 0 and b2 satisfy the following condition: there exists a constant y > 0 such that for any x1,x2 E Z with X = max(]xr 1,1x21) suficiently large one has

I

x1 +x2-

bz I bl

> emTX21nX .

Then the field L3 has transcendence degree at least 2. Applying Theorem 6.1 to Gel’fond’s third conjecture with ai = /Ii-l and bi = pi-l In a, i = 1,2,3, we seethat if d 2 3, then at least two of the numbers afl, &, & cup* are algebraically independent over Q. Here the condition (2) holds by Liouville’s theorem, since p is algebraic. In particular, this means that tr deg Q ( cyp,a@, . . . , oPdel) 2 2 for d>3. (4)

262

Chapter

6. Values of Functions

That

Have an Addition

Law

When d = 3, one obtains the following result, for which Gel’fond had published a direct proof somewhat earlier (see [1949b]). Theorem 6.3. Let cr be an algebraic number not equal to 0 or 1, and let p be a cubic irrationality. Then

are algebraically independent over Q. As Gel’fond pointed out in [1949d], Theorems 6.1 and 6.2 imply that each of the following sets contains at least two numbers that are algebraically independent over Q (here v E Q, v # 0, and a E A, a # 0,l): 1) e, eeY, ee2”, eeSY; 2) e, ueY , ae2”, a”“, a’*” ; 3) In a, aln” a, &12” a, &IS” 0.

1.2. Bound for the Transcendence Measure. Gel’fond’s method of proving algebraic independence was based on a new technique for estimating the transcendence measure. As an indication of the importance of such estimates, Gel’fond gave a proof of the following theorem along with the proofs of Theorems 6.1 and 6.2 in [1949d]. Theorem 6.4. Let a, b, (Y,p E A, a # 0, 1, and supposethat b and In (.y/ In fi are irrational. Further suppose that P(x) E Z[x], deg P = p > 0, H(P) = H 2 1. Then for any fixed E > 0 and H > Ho(e) one has -&(P+ln

IW)l and

fJ)(qP+ln

>e

w)2+r

I( >I p

$

>

e-P2(P+m2+=

.

In these inequalities for the first time p = deg P and H = H(P) can vary independently of one another. In particular, the bounds hold for fixed H and increasing p (compare with Theorems 3.19 and 3.20). We briefly describe the idea behind Gel’fond’s estimate for the transcendence measure. Let P(x),Q(x) E Z[x] be two polynomials with no roots in common. Then there exists a nonzero integer R and two polyomials A(x),B(x) E Z[x] such that P(x)A(x)

+ Q(x)B(x)

= R.

(5)

Here R is the resultant of P(x) and Q(z); and the coefficients of A(x) and B(x) can be expressed in terms of the coefficients of P(x) and Q(x). Using these expressions, one derives bounds on the degrees and coefficients of A(x) and B(x) in terms of the degrees and coefficients of P(x) and Q(x). From

$1. Gel’fond’s Method and Results

263

these bounds (and the fact that ]R] > 1) we obtain the following lemma as a consequence of (5). Lemma 6.1. Suppose that w E C!, and the polynomials P(z), Q(z) E Z[z] have no roots in common. Then

m4W411 IQ64 L 03+ ~)-lII~II-qIIQII-P , where p = deg P, q = deg Q, lIPI is the length of the vector of coeficients of P, and ~~Q~~ is the length of the vector of coeficients of Q. This lemma can be used to obtain a bound for the transcendence messure of w, i.e., a lower bound for IQ(w)] in terms of degQ and H(Q), as follows. Suppose that for given Q(z) we have somehow managed to construct a polynomial P(z) whose a) b) c)

roots are distinct from the roots of Q(z); degree deg P and height H(P) are not very different from the degree and height of Q(z); value at w is fairly small, namely, IP( 5 IQ(w)l.

Then the inequality in Lemma 6.1 implies the bound

IQ(w)I 2 (P+ d-lIIPII-qIIQII-P 7 which, by property b), can be expressed in terms of just q = deg Q and H(Q). It is interesting to note that the analytic construction of P(z) that, Gel’fond used in his proof of Theorem 6.4 was similar to the construction of the auxiliary function in the solution to Hilbert’s seventh problem. lJe obtained a polynomial P(z) whose coefficients are not in Z, but rather in the ring of integers Zk of a certain algebraic extension of Q. Then Gel’fond used a lemma analogous to Lemma 6.1 for P(z), Q(z) E &[z]. The above version of Lemma 6.1, which is an improvement of Gel’fond’s original lemma in the case k = Q, is taken from [Brownawell 19741. Another important consideration is that in the analytic construction of P(x) one cannot be assured that condition a) holds. For this reason one imposes an additional restriction on Q(z): Q( x ) must be an irreducible polynomial. Condition a) can then be replaced by the weaker condition a’) P(x) is not divisible by Q(z). Since one can ensure that condition a’) holds in the analytic construction of P(z), in this way one can prove the lower bound for IQ(w)1 for irreducible polynomials Q(z). To obtain a bound for arbitrary Q(x), one usesthe following lemma of Gel’fond. Lemma 6.2. Let &I (21,. . . , xm), . . . , Qs(zl,. . . , x,) be arbitrary polynomials incC[zi,... ,x,1. Let ni, i = 1,. . . ,m, denote the degree of

Qh,.

. . ,4

= &1(~1,...,xm>...Q~(x1,...,

x,)

264

Chapter 6. Values of Functions That Have an Addition Law

in the variable xi, and set n = nl + . . . + ns. Then H(Q)

2 ewn H(Qd

. . . H(Qs)

.

Lemma 6.2 is used repeatedly in the proof of Theorem 6.4. We shall describe only one of the circumstances when it is used. If a lower bound has already been proved for the value at w of irreducible polynomials, then one can apply the lemma to obtain a bound for IQ(w)] f or arbitrary Q(x). To do this, one writes Q(x) as a product of irreducible factors, for each of which one has a lower bound for the value at w. This leads to a lower bound for IQ(w) 1, but one that is expressed in terms of the degrees and heights of the irreducible factors of Q(x). If we now use Lemma 6.2 with m = 1, along with the fact that each irreducible factor has degree 5 deg Q(z), we can express our estimate in terms of the degree and height of Q(x). This gives the required result. Finally, we note that lemmas similar to Lemmas 6.1 and 6.2 (with m = 1) were first proved and the above method of bounding polynomials was first used in [Koksma and Popken 19321 to obtain a result on the transcendence measure of en. In 1948, Gel’fond was the first to apply this argument to prove algebraic independence of numbers. In what follows we shall give an outline of the analytic construction of the polynomials P(x) satisfying a’), b), and c).. 1.3. Gel’fond’s “Algebraic Independence Criterion” and the Plan of Proof of Theorem 6.3. We now give a brief description of Gel’fond’s technique for proving Theorem 6.3. We shall make use of some technical simplifications and refinements that were found later. The next lemma, which concerns dense sequences of polynomials, provides the algebraic basis for the proof of Theorem 6.3. The lemma is sometimes called Gel’fond’s “algebraic independence criterion” for reasons that will soon become clear. Lemma 6.3. Suppose that w E Cc, and a(N) zs . a real-valued function that is defined for all N E N suficiently large, is monotonic increasing, approaches infinity as N + 00, and has the property that a(N + 1)/c(N) is bounded. Further suppose that there exists a sequence of polynomials PN(x) E Z[x], PN $0, such that t(PN>

5 a(N),

where c is a suficiently large constant. &T(W) = 0 for all N suficiently large.

IpN(w)l

< e-

co’(N)

,

Then w is an algebraic

In Gel’fond’s version of this lemma, in that increases monotonically to +oo. The proved in [Lang 19651. The proof uses the A simple consequence of Lemma 6.3, Gei’fond, was essentially what was used

number,

and

place of c one has a function O(N) above form of the lemma was first ideas that were described in $1.2. although not stated explicitly by in his proof of Theorem 6.3. It is

51. Gel’fond’s Method and Results because of this corollary that Lemma 6.3 is sometimes gebraic independence criterion.”

265 called Gel’fond’s

“al-

Corollary 6.1. Suppose that ~1, ~2, . . . , wm E Cc, m 2 2, and the function cl(N) satisfies the same conditions that o(N) did in Lemma 6.3. Further suppose that there exists a sequence of polynomials QN(x~ , x2,. . . , x,) E ,x,1 that satisfy the following conditions for any constant cl > 0 q7a,x2,... and any suficiently large N E N:

t(QN) I

m(N),

0<

~QN(wI, ~2,. . . ,wm)l I e- clu:(N) .

Then there are at least two of the numbers ~1, ~2, . . . , WN that are algebraically independent over Q. We show how this corollary follows from Lemma 6.3. If all of the numbers wi,...,w, were algebraic, then, using Liouville’s theorem to bound from below, we would easily arrive at a contradiction. IQN(wI,. . . , w,)] Thus, there must be a transcendental number among the wi, and without loss of generality we may assume that wi is transcendental. If the corollary were false, then all of the numbers QN(w~, . . . , w,) would be algebraic over the field Q(wi), and for some fixed a E Z[wi] the numbers (N = adegQN QN(w~, ... . , w,) would be algebraic integers over the ring Z[wi]: This implies that the norm from Q(wi, . . . ,w,) to Q(wi) of tN belongs to the ring z[wl], i.e., it is the value at wi of some polynomial pN(z) E z[x]. It is easy to see that PN(x) satisfies the conditions in Lemma 6.3 with a function U(N) that differs from gi(N) only by a constant factor. But then Lemma 6.3 leads to a contradiction, since, by construction, all of the PN(x) are nonzero at wi. Thus, the numbers wz, . . . , wm cannot all be algebraic over Q(wi), and the corollary is proved. The rest of the proof of Theorem 6.3 basically consists of an analytic construction of a sequence of polynomials QN(x~ , x2, x3, x4, xs) E Z[xi, x2, x3, x4, x5] that satisfies the conditions in the corollary with WI = op, wg = CP, w5 = (Y. Since deg /3 = 3, it follows that ws, ~4, and ws w3 ,a@ ) w4 = CR, are algebraic over Q(wi , ~2). Hence, in order for there to be two algebraically independent elements among wi , . . . , ws, the numbers wi = ~0 and ws = a@ must be algebraically independent over Q. To construct the required sequence of polynomials one uses much the same procedure as in the solution of Hilbert’s seventh problem (see Chapter 3). First, we construct an analytic function F(z) depending on a parameter N:

where

AwczkaE Q4, and

t(&kzks)

6 3N3i2(1n N)lj4 ,

266

Chapter 6. Values of Functions That Have an Addition Law

F(qll for all integers

+ 12p+ /3P2) = 0

s, 11, 12,/s in the intervals

0 5 li 5 L = [N”2(lnN)“4]

0< s 5 s =

)

1

XiNs/2(lnN)-s/4

)

where [ ] is the greatest integer function and Xi is a sufficiently small positive constant. The next step in the proof uses a bound on the number of zeros of F(z) (in place of the interpolation procedure that was used in the solution of Hilbert’s seventh problem). Gel’fond’s lemma giving this bound depends on some arithmetic conditions that in the present situation are automatically fulfilled, but in general (Theorems 6.1 and 6.2) require that the additional restrictions (2) and (3) be imposed on the numbers. In [1971a] Tijdeman proved an estimate without needing those conditions, and this made it possible to remove the restrictions (2) and (3) from the theorems (see 51.4 below). Lemma6.4. Supposethatn,mo,ml,..., m,-1 EN,m=mo+...+m,-1, (~~-1 are distinct complex numbers, a = max, 1~~1, and the function E(z) is defined by setting

~o,W,...,

n-lm,-1

E(z)

= c c APvzPeav*, v=o p=o

Then the number R is at most

of zeros of E(z)

( counting

multiplicity)

in a disc of radius,

3m+4aR. Tijdeman’s

lemma implies that there exist integers

0 5 1: 5 N1/2(lnN)1/4, Fql;

s’, Zi, Zk, 1; such that

0 5 s’ 5 4N312(ln N)-3/4

,

+ 1;p + Zg12) # 0 .

It is easy to see that there exists a polynomial QN E Z[xi, x2, x3, x4, z.E,~such that QN(WI,W~,WQ,W~,W~) = (lna)-8’F(s’)(1; +$p+l;p2) and t(QN) But from the representations

and

5 15N3/2(1nN)1/4

,

51. Gel’fond’s

Method

and Results

267

where zi = Zi + I!# + lip”, r is the circle ]
< e--3X1NS’nN .

= Thus, wl,... , ws satisfy the conditions in Corollary 6.1 with or(N) = 15N3/2(1nN)1/4; hence, wr = ofl and w2 = a@ are algebraically independent over Q. This completes the proof of Theorem 6.3. To prove Theorem 6.4 one uses roughly the same analytic construction, except that in place of Lemma 6.4 one needs a subtler result. Namely, one has to have a bound for the maximum coefficient A,, of E(z) in terms of the maximum value E = ma+,,, I&)(/3,,)] of the derivatives of E(z) on a certain set of points, i.e., one needs an inequality of the form max IAPVl 5 cE, where the constant c > 0 can be written explicitly in terms of parameters that describe the location of the points ,& and a, on the complex plane (see [Tijdeman 1973a]). 1.4. Further Development of Gel’fond’s Method. In [1968bJ Shmelev used Gel’fond’s method to prove an analogue of Theorem 6.1 in the case p = 4 and q = 2. In particular, he obtained the following corollary (which he also proved in [1968a]): Suppose that oi and CQ are algebraic numbers whose logarithms are linearly independent over Q, p is a quadratic irrationality, and Then at least two of the three numbers 77= lno2/lnai.

are algebraically independent over Q. In [1971b] Tijdeman used Lemma 6.4 and, changing very little in Gel’fond’s method, was able to prove that in order for Theorems 6.1 and 6.2 and Shmelev’s theorem to hold all one needs is that al,. . . , ap and bl, . . . , b, be linearly independent over Q. Namely, if this condition is fulfilled, then: trdegL2

> 2

for

p=q=3;

(6)

trdegL2

2 2

for

p=4,

q=2;

(7)

trdegLs

2 2

for

p=3,

q=2.

(8)

All of the results proved using these ideas can be combined in the following theorem. Theorem 6.5. Let al,. . . , aP and bl, . . . , b, be two sets of complex numbers that are linearly independent over Q. Then: I) 2) 3)

if pq 2 2(p + q), then tr deg L1 2 2; if pq > p + 2q, then tr deg L2 > 2; ifpq >p+q, thentrdegLs 2 2.

Part 1) of Theorem 6.5 was proved in [Shmelev 19711with an additional technical condition similar to (2) ( a condition that automatically holds if

268

Chapter 6. Values of Functions That Have an Addition

Law

the ai and bj are real). An unconditional proof using Lemma 6.5 was given independently by Waldschmidt [1971] and Brownawell [1972]. Part 2) of the theorem follows from Tijdeman’s inequalities (6) and (7), and it was also published by Waldschmidt [1971], who had independently proved a result similar to Lemma 6.4. Part 3) follows from Tijdeman’s inequality (8) and was also proved in [Waldschmidt 19711. For a short proof of this result without using any lemma similar to Lemma 6.4, see Theorem 3 of [Shmelev 19711. Different proofs of Lemma 6.3, as well as refinements and modifications of the lemma, can be found in many places. For a list of references, see [WaldSchmidt 1974c]. Brownawell [1975] and Waldschmidt [1973a] independently proved a generalization of Lemma 6.3 in which deg PN and In II are bounded by different functions 6(N) and o(N). They used this result to prove the following theorem. Theorem 6.6. Suppose that al, a2 and bl, bz are two pairs of complex numbers each of which is linearly independent over Q. If

are algebraic numbers, then at least two of the six numbers al,

a2, bl,

b2, ealbz, ew&z

are algebraically independent. If we choose al = b2 = 1 and a2 = bl = er for some nonzero r E Q, then this theorem implies that either ee’ or ee2’ must be transcendental. In the case r = 1 this proves a conjecture of Schneider (see p. 138 of [Schneider 19571). Another consequence of the theorem is the following: if e”* E A, then e and T are algebraically independent. To seethat, take al = bl = ni and a2 = b:! = 1. The above method has also been used by Chudnovsky to prove bounds for transcendence measures of various numbers. 1.5. Fields of Finite Transcendence Type. The discreteness of Z has been very important in proving the results in this section. The analogous fact for algebraic numbers is Liouville’s theorem giving a lower bound for the modulus of a nonzero algebraic integer in terms of the degree and height of the number. In [1966] Lang proposed an inductive procedure for bounding the transcendence degree of finitely generated fields. His method is based on the notion of.a field of finite transcendence type. Let K = Q(wr , . . . , wq, C) be a finitely generated field extension of Q, where wi , . . . , wq are algebraically independent over Q and C is algebraic of degree v over Q(wi , . . . , We). Any cy E K can be written in the form

Cl= where Pi E Z[sl,...,

Pa@) + Pz(iq

x~],z=

+ .** + P”(qc”-l PO(4

(wl ,...,

up).

,

$1. Gel’fond’s

Method

and Results

269

Definition 6.1. By the size of o, denoted s(a), we mean the smallest B such that there exists a representation (9) with t(Pi) 5 B for i = 0, 1, . . . , V. Following Lang, for r 2 2 we shall say that IK is a field of transcendence type at most r if K has a set of generators ~1,. . . , wq, < relative to which every nonzero Q E lK satisfies the inequality

I4 2 exp C---c4~)~)

(10)

for some constant c that depends only on WI,. . . , wp, C. In that case we write T(K) 5 7. In [Waldschmidt 1974c] it is proved (see Lemma 4.2.23) that bounds on the transcendence type of a field do not depend on the choice of generators. If we have a sufficiently good bound on the algebraic independence measure for some transcendence basis of a field, then we obtain an upper bound for the transcendence type of the field. It is easy to verify that Gel’fond’s Theorem 6.4 implies that, under the conditions in the theorem, 7- (Q(ab)> I4

and

r(Q($))

14+e

for any E > 0. And from Theorem 3.33, which gives a bound for the transcendence measure of X, we obtain r(Q(r)) 5 2 + E for any E > 0. In [Cijsouw 1972, 19741one can find bounds on the transcendence measure of other numbers, from which one obtains upper bounds for the transcendence type of the corresponding fields. Using the Dirichlet pigeonhole principle, it is not hard to prove that the transcendence type of a field K cannot be less than 1 + tr deg@. The inequality (10) can be used in place of Liouville’s theorem to prove that numbers are transcendental or algebraically independent over the field K. If one is able to show that there are at least m of the numbers 51,. . . , J,, that are algebraically independent over K, then this means that there are at least m+qofthenumberswi ,..., wp,Ji ,..., & that are algebraically independent over Q. As an illustration of what can be obtained in this direction, we give the following result (see Part 4 of [Waldschmidt 1974c]). Theorem 6.7. Let T > 1 be a real number, and let K be a finitely generated subfield of @ having transcendence type at most r. Suppose that al, . . . , aP and b are sets of complex numbers that are both linearly independent over tri-y

* Pl>dP+d

9

then at least one of the numbers eaibj, 1 5 i 5 p, 1 2 j 5 q, is transcendental over IK.

In the special case when lK = Q(wi) with wi transcendental, this theorem is proved in Chapter 5 of [Lang 19661. Similar results for the numbers

270

Chapter 6. Values of Functions That Have an Addition

Law

{ai, ea;bj}, 1 I i 5 p, 1 5 j 5 q, for (pq +p)/(p

+ q) 2 r and for the numbers {Ui, }, 1 I i 5 p, 1 I j 5 q, for (pq + p + q)/(p + q) 2 7 were announced in [Brownawell1972] and [Waldschmidt 19711. Proofs can be found in [Waldschmidt 1973b]. There are more general results of this type in [Shmelev 1975a]. .One also has analogues of Theorem 6.5 that hold for fields of finite transcendence type. These theorems can be proved by carrying over Lemmas 6.1-6.3 to such fields (see [Shmelev 19741 and [Brownawell 19751). The next theorem is taken from [Brownawell 19751. bj,eci*bj

Theorem 6.8. Suppose that r(K) 5 7, and both al,. . . ,ap and br,. . . , b, are linearly independent over Q. Then: I) ifpq/(p + q) > 27, then tr deg,lK(e”‘bj) 2 2; 2) if (pq + p)/(p + q) > 27, then tr deg,lK(oi, eaibj) 2 2; 31 if(pq+p+q)/(p+q) 2 2r, thentrdegK~(ai,bj,eaibj)

2 2.

The first example of a field of transcendence degree 2 having finite transcendence type was found in [1982a] by Chudnovsky, who proved the following: if the Weierstrass function p(z) has algebraic invariants g2 and 93, complex multiplication, and period lattice R, and if w E 0\20 and q is the correspond‘ing quasiperiod, then for any E > 0 the transcendence type of Q(x/w, v/w) is at most 3 + E. We shall give an estimate for the algebraic independence measure of r/w and q/w in $5 (see Theorem 6.38). In this case the above bound on the transcendence type differs only by E from the best possible value of 3. The first (and apparently the only) example of classes of numbers that generate fields of transcendence degree greater than 2 and finite transcendence type were given in [Becker 1991a] and [Nishioka 1991a]. Those authors proved good bounds depending on H(P) and deg P for the algebraic independence measure of the values at algebraic points of functions that satisfy the so-called Mahler functional equations. The transcendence and algebraic independence of the values of such functions were first studied by Mahler in [1929, 1930a, 1930b]. Here is one of the results that follow from Theorem 2 of [Nishioka 1991a]: Let d E N, d 2 2, and let p(z) be defined by the series

cp(.z)= 2 Zd” . n=O Then for any Q E A, 0 < 1~11< 1, and for any E > 0 one has -r (QM~,Q~~~>, This upper bound lower bound. The uses ideas that will Finally, we note the transcendence

. . . , dad-‘I)

L d + 1+ E.

on the transcendence type is rather close to our earlier method of proof in [Becker 1991a] and [Nishioka 1991a] be described later (see s3.3). that for almost all w E Iw in the sense of Lebesgue measure, type of Q( w ) is less than or equal to 2 (see [Nesterenko

32. Successive

Elimination

of Variables

271

1974a]). An analogous result holds for almost all complex numbers w (in the senseof Lebesgue measure in C). For m 2 2 the transcendence type of fields of the form Q(wi, . . . ,w,)ClRis~m+2foralmostall(wi,...,w,)EBm (see [Nesterenko 1974a]). It has been conjectured that the transcendence type ,w,) c lR is 5 m+l for almost all (wi,...,w,) E R”. Here OfQ(Wl,... “almost all” is meant in the senseof Lebesgue measure in I+?. In [1990, 19941 Amoroso proved that for almost all points in Cc” the transcendence type of the corresponding field is 5 m + 1. The proof usesideas that will be discussed in 53.

$2. Successive Elimination

of Variables

In this section we describe improvements of Gel’fond’s method that under certain conditions enable one to prove that some fields generated by values of the exponential function have transcendence degree at least 3, 4, etc. The techniques used all involve repeated application of results similar to Lemma 6.1. 2.1. SmaIl Bounds on the Transcendence Degree. In [1972a] Shmelev published the first result giving three algebraically independent numbers among the values of the exponential function at transcendental points. Namely, he was interested in finding a lower bound for the transcendence degree of the field where Q E A, a # 0, 1, and p is an algebraic number of degree d. Shmelev proved that then tr deg K 2 3 . d 1 19, if (11) His argument follows the plan of proof of Theorem 6.4. The role of Z is played by the ring Z [&I, and Liouville’s theorem is replaced by the bound in Theorem 6.4 for the transcendence measure of op. Thus, his proof is an example of an argument using the transcendence type, as described at the end of the last section. In [1972] Brownawell announced several results on algebraic independence of values of the exponential function. In [1975] he published a proof of the following result, which strengthens Shmelev’s theorem:

if

d > 15,

then

tr deg lK 13 .

(12)

A further improvement on (11) and (12) was given by Chudnovsky [1974a]: if

d 2 7,

then

trdeglKz3.

(13)

Then in [1974/1975] Waldschmidt refined Chudnovsky’s argument and proved that

Chapter 6. Values of Functions That Have an Addition Law

272

if

h e improved

and in [1975/1976]

then

d 2 31,

if

trdegK24;

this result as follows:

d 2 23,

then

trdegK

2 4.

Other expository treatments of this approach can be found in [Waldschmidt 1977a] and [Brownawell 1979b]. In this connection we should also mention [Chudnovsky 1977a, 1977b], [B rownawell1975/1976,1979a], [Brownawell and Waldschmidt 19771, and [Mignotte 1974/1975]. 2.2. An Inductive Procedure. The identity (5), which lies at the heart of the proof of Lemma 6.1, holds for polynomials in one variable over an arbitrary ring. This means that it can be applied to polynomials P, Q E , z,], regarded as polynomials in 21 with coefficients in the ring Z[z1,22,... x,]. Then the resultant, which we denote R = R(x2,. . . ,x,) = qx2,..., Res,, (P, Q), is a polynomial in Z[xa, . . . , xm]. By analogy with Lemma 6.1, one has the following inequalities for any m-tuple of complex numbers Wl,...,W,Z t(R) . . , w,)l

IR(w2,.

clt(WQ)

5

7

5 eczt(P)t(Q)max(lP(wl,w2,.

. . ,wm)],

(14)

IQh,~2,-~-,an)I)

1

where cl > 0 and c2 > 0 depend only on m, wi, . . . , w,. Of course, there is a condition that must be satisfied in order for P and Q to have nonzero resultant - a condition that is also sufficient for (14) to hold. Namely, the polynomials P(Xl,X2,*.. ,x,> and Q(a,m . . . ,x,) cannot have any common divisors in xm] in which the variable xi appears. qn,x2, f.. , We now consider a hypothetical procedure for bounding the transcendence degree of a field K. Let wi, ws, . . . , w, E Cc,m 2 1, be a transcendence basis for K, so that m = tr deg K. Suppose that we have somehow been able to construct a set of polynomials PI,. . . , P, E iZ[xl, x2,. . . , z,,J, s > m, such that t(P,> I T,

. , w,)l

IPi(W,W2,~~

<

i=l,...,s,

emu,

and also the following elimination of variables procedure can be carried out. We set P.(l) = Res51(PI, Pi) E Z[xz, “‘7 x,], pi2) z = Resx.2 (p(l) 2 ) P,“‘, . p!“-‘1 z

.

.

. =

. Res

.

.

z,-l

.

.

E Z[x, .

.

.

xm], i = 3,. . . ) s ;

,.“7 .

.

i = 2,. . . ) s ;

.

.

.

.

(PiTy2), Pjmm2’) E Z[x:,],

.

.

.

.

.

.

.

i = m, . . . , s .

.

.

.

.

.

.

$2. Successive Elimination

of Variables

273

We assume, first, that all of the above resultants are nonzero; and, second, that among the polynomials P!“-“(z,) E Z[zm], i = m >“‘, s, there exist two that have no common root:. Applying (14) m - 1 times, we find that t (P/“-“) IP!“-“(w,)l 2

2 CATTY-‘, 5 exp (-U

+ CATTY-‘)

i = m, . . . , s , i = m,. . . ,s .

,

Here and later, the letter c with various subscripts denotes positive constants that depend only on m, WI,. . . , wm. Finally, if we apply Lemma 6.1 to the two polynomials PJ”-l) (z,) that have no common roots, we obtain the inequality z exp (-U

+ CATTY) > exp (-csT~~)

,

from which it follows that U 5 CATTY. This inequality gives a lower bound for m = tr deg K. When bounding the transcendence degrees of the fields Li, it should be noted that Gel’fond’s construction, which was used by Tijdeman, Shmelev, Waldschmidt, Brownawell, and Chudnovsky, enables one to obtain a sequence of polynomials P~(z1, . . . , z,) with parameters T M NP+Q, T M Np+q,

U

T M Nf’+q,

U x NPqInN w

NP9+P(ln u

N)P/(P+‘d x

NPQ+P+Q

in the case of LI , in the case of L2 ,

(15)

in the case of L3

(see Theorems 7.1.6, 7.2.8, and 7.3.4 of [Waldschmidt 1974c]). In the case of LI, for example, this line of argument should give the bound m 2 l+ [log2 (iz)] . However, in general this set of polynomials does not have the properties needed to eliminate the variables. It turns out that it is a complicated matter to ensure that the conditions hold for all of the steps in the elimination of variables. The results in $2.1 come from an implementation of these ideas when the number m of variables is small. The first attempt to carry out this approach in the general case (see Theorem 6.10 below) was due to Chudnovsky [1974d, 1974e]. Although he was not able to give a complete proof of the results he announced (he published a proof only in the “nondegenerate case,” i.e., when one essentially assumes that all of the steps go through in the elimination of variables), nevertheless the preprints [Chudnovsky 1974d, 1974e] played an important role as a stimulus that inspired many researchers to look for a complete proof of these and similar results based on Chudnovsky’s ideas. The difficulties encountered in carrying out this plan of proof caused people to pay special attention to the algebraic problems connected with elimination theory, and to look for conditions on the original set of polynomials Pi E

274

Chapter 6. Values of Functions That Have an Addition

Law

1 5 i 5 s, that are as simple as possible and yet ensure . . , z,], that all of the successive steps in the elimination can be realized. With this purpose in mind, attempts were made to generalize Lemma 6.5 (Gel’fond’s algebraic independence criterion) using the techniques described above. The direct generalization of Corollary 6.1 of Lemma 6.3, with (61(N))& rather than ((TV) 2 in the exponent of e in the hypothesis and with the phrase “there are at least K of the numbers” in the conclusion, turns out to be false (see Theorem 14 in Chapter 5 of [Cassels 19571 for a counterexample). The first attempt at a correct generalization of Lemma 6.3 was made in [Dvornichich 19781, but that result was never applied to prove algebraic independence of numbers. Subsequently, several generalizations of Lemma 6.3 appeared that were suitable for proving algebraic independence. The first was due to Reyssat [1981]. The paper [Waldschmidt and Zhu Yao Chen 19831 should also be mentioned. We shall state a result that can be derived from [Nesterenko 1983b] and is a slight generalization of the theorem in [Reyssat 19811. q21,22,.

Theorem 6.9. Suppose that ~1, . . . , w,,, E Cc; (~1(N) and 02(N) are nonnegative functions of a natural number argument such that ol(N + 1) 5 sol(N) for some a 2 1; K > 1; and the sequence of polynomials PN E Z[xl, . . . , xm], ‘N > No, satisfies the conditions

-N”ol(N)

< In ~PN(w~, . . . ,wm)l

5 -N”czz(N)

.

Then there exists a constant c, depending only on WI,. . . , w, and K, such that a2(Nj2

1

cm(N)

all N 2 No, then there are at least 1 + [log, K] of the numbers WI, . . . , w, that are algebraically independent over Q.

for

In [1981] Philippon used a padic version of the criterion in [Reyssat 19811 to prove that if certain bounds hold for linear forms in the p-adic numbers al,..., ap and bl,. ..,bg, then PC7 tr deg L1 2 log, p+q

Pq+P tr deg L2 2 log, P+q

’

Pq+P+q P+q * In particular, if (Y # 0,l is algebraic and /3 is an algebraic number of degree d 2 2, then there are at least log,(d + 2) - 1 of the padic numbers tr deg LI 2 log,

lna,

c#,

#,

. . . , QIBd-’

that are algebraically independent over Q.

92. Successive

Elimination

of Variables

275

There are some difficulties in applying Reyssat’s criterion with Gel’fond’s construction, as was done, for example, by Chudnovsky when he proved (13). In [1981] Philippon proposed an analytic method involving functions of several complex variables (“false” variables) to construct the polynomials PN E %[xl,... , zm] that are needed to apply the criterion. To prove the bound on the transcendence degree of Ns, Philippon constructs the following auxiliary function F(Z) of d complex variables f = zd) (we have slightly modified Philippon’s construction to make it (G,..., better suited to the complex case): (16) _Here4_hesummat@ is over all vectors Ji = (pi,. . . , pd), 0 5 pi 5 No, and x = (A,,... ,Ad), , Xi,), 0 5 Xij < N, with integer components Ai = (kl,... and with parameters N and NO, where No depends on N in some explicit way. The vector Z in (16) has the form h = (al,. . . , up). The coefficients C(x, Ti> are chosen to be rational integers, and it is this that distinguishes Philippon’s construction from Gel’fond’s; in particular, Philippon can use the trivial lower bound maxIF ]C(x, Ti>] 1 1. Recall that in Gel’fond’s construction the coefficients were’ chosen to be polynomials in the numbers being studied with integer coefficients, and it was not easy to find a lower bound for the maximum coefficient. The .coefficients C(x, JX)E Z are chosen so that the function F(Z) has zeros of multiplicity at least K at the points hibi + . . . + h,b, for 0 5 hj < H, where H and K are parameters that are chosen as a function of N. F(Z) can be represented as an exponential polynomial in zd whose coefficients are functions of a similar type in the variables zi , . . . , z&i. This makes it possible to use analytic results for functions of one variable (the Hermite interpolation formula, and Tijdeman’s lemma giving an upper bound for the coefficients of an exponential polynomial) along with induction on the number of variables. If the number d of variables is chosen greater than a certain bound that depends only on p and q, then it turns out that there exist parameters Lo, H, and K for which one can construct polynomials satisfying the conditions in Reyssat’s criterion with upper bound only a little worse than (15).

In [1984] Endell published a brief outline of a direct proof of the following theorem, which had been announced earlier by Chudnovsky (see [Chudnovsky 1974d, 1974e, 1977, 1982c]). Theorem 6.10. Let al,. . . , up and bl, . . . , b, be complex numbers for some constant y > 0

and

lwbl + . . * + vqb,l > exp(-y]??])

such that

276

for

anyE= Then

Chapter

(Ul,...

6. Values of Functions

,up) E zp a7dv=

That

Have an Addition

Law

(WI,... ) wq) E zq WJith IEl > 0, Iv’1 > 0.

1) ifpq/O,+q) 2 2”, thentrdegLi >n+l; 21 if(pq++)/(p+q) >2n, thentrdegL2 >n+l; 3) if(pq+p+q)/(p+q) >2”, thentrdegLs >n+l. The proof of the algebraic independence criterion in [Nesterenko 1983c] was based on ideas that will be described in the next section. Using the same ideas, in [Nesterenko 1983a] it was proved that trdeg~(cuP,aprr,...,apd-I)

2 [log,(d+l)],

(17)

where Q # 0,l is an algebraic number, and /3 is an algebraic number of degree d 2 3. It is easy to see that this result is a special case of part 2) of Theorem 6.10.

§3. Applications

of General

Elimination

Theory

3.1. Definitions and Basic Facts. The proofs in [Nesterenko 1983a, 1983cj of the algebraic independence criterion and the inequality (17) were based on the idea of working with ideals of the ring Z[ze, . . . , z,] in the same way as with polynomials. The basic definitions and facts needed for this section can be found in [Zariski and Samuel 19601. Here we shall only give a few of the definitions. We first define the rank of an ideal. (In [Zariski and Samuel 19601this is called the height, but we prefer the term “rank,” because “height” has a very different meaning in transcendental number theory.) Definition 6.2. The rank h(p) of a prime ideal p c Z[ze, . . . , z,] is the maximum length of an increasing chain of prime ideals that are strictly contained in p. If J is an arbitrary ideal, then its rank h(3) is defined as the minimal rank of the prime ideals containing 3. Note that principal ideals have rank 1. Recall that any ideal 3 c Z[se, . . . . . . 7z,] has finitely many associated prime ideals pi, . . . , pB (these are analogues of the prime divisors of an integer). Definition 6.3. An ideal J c Z[zs, . . . , zrm] is said to be unmixed if all of its associated prime ideals pj have the same rank h(3). In the case of an unmixed ideal 3 the set of associated prime ideals is uniquely determined. In [Nesterenko 1983a, 1983c] certain numerical invariants deg3 and H(3) - analogous to the degree and height of a polynomial P E Z[ze, . . . , zm] were associated to any unmixed ideal J c Z[ss, . . . , z,] that is homogeneous (i.e., generated by homogeneous polynomials). For any CJ = (we,. . . , w,) E

$3. Applications

of General

Elimination

Theory

277

@m+1 \ {0}, the magnitude of the ideal at Z, denoted ]J(ij)], was also defined;

this is analogous to the absolute value of P(Z) for a polynomial P. The precise definitions of deg J, H(J), and ]J(Ts)] are based on the elimination theoretic notion of the u-resultant or Chow form of the ideal J. In the case of homogeneous prime ideals of the ring Ic[zc, . . . ,zm], where Ic is a field, this object was introduced in [Chow and van der Waerden 19371,where it was called the “zugeordnete Form.” It is the resultant of a set of basis polynomials for 3 and a certain set of linear forms with indeterminate coefficients. The properties of the Chow form are explained in detail in [Hodge and Pedoe 19471, where it is called the “Cayley form.” The name “Chow form” goes back to 1972, when the Russian edition of [Shafarevich 19941 was published. It should be noted that the non-homogeneous analogue of the Chow form had appeared earlier in articles by Hentzelt [1923] and Noether [1923], where it was called the “Elementarteilerform.” Its properties were subsequently studied by Krull [1948], who referred to the ‘LGrundpolynom.” We also note van der Waerden’s paper [1958]. In these classical papers the Chow form was studied from an algebrageometric point of view. The behavior of the coefficients, which is what one has to understand in order to apply it in transcendental number theory, was not examined, although essentially all of the necessary tools for this had been developed. The invariants deg J, H(3), and ID@)] of an unmixed homogeneous ideal J that were introduced in [Nesterenko 1983a, 1983c] have properties similar to those of the degree and height of a polynomial and the value of a polynomial at a point. If J = (P) C Z[ZO,. . . ,z,] is a principal ideal, then one has (see Proposition 1 of [Nesterenko 1984b]): deg J = deg P,

In H(J) 5 In H(P) + 2m2 deg P ,

We now consider the following problem: Given a point z = (us,. . . , w,) E Cm+‘, find a lower bound for 13(w)I in terms of H(3) and deg3. In the case of a principal ideal J = (P) it follows from (18) that a bound for ]J(ss)] leads to a lower bound for IP( in terms of H(P) and deg P. Results analogous to Lemmas 6.1-6.3 hold for these invariants of ideals of Z[zs, . . . , zm]. The numbers deg 3, In H(3), and In ]YJ(z)] behave “almost linearly” when the ideal is factored into an intersection of primary ideals. This makes it possible to reduce the problem of estimating ]J(~s)] for an arbitrary unmixed ideal 3 to that of estimating In@)] for a prime ideal p. An analogue of Lemma 6.1 enables one to construct an inductive argument (with induction on the rank of 3) establishing a bound for ]?(Ts)]. Given a prime ideal p and a polynomial Q E Z[zc, . . . , z,] \p, this lemma gives an ideal 3 with h(3) > h(p) such that H(3), deg3, and ]3@)] can be bounded from above in terms of the corresponding invariants of p and Q. In essence,the lemma replaces the

278

Chapter 6. Values of Functions That Have an Addition Law

procedure of elimination of variables. The basic idea of [Nesterenko 1983a, 1983c] was to use induction on the rank of the ideals in place of the elimination of variables. In [1984, 1985, 19861 Philippon made a major advance in the theory that enabled him to strengthen all of the bounds on the transcendence degrees of the number fields, by replacing logarithmic bounds with linear ones (see Theorem 6.12 below). Philippon refined the analogue of Lemma 6.1 by introducing a new invariant (the projective distance p from the point i5 to the variety of zeros of the ideal) and finding a connection between 13(Z)] and the distance p. In [Philippon 1984, 1985, 19861 the invariants of ideals are determined for the polynomial ring k[zc, . . . , z,] over a finite extension k of Q; in particular, he uses the Weil height of a point in projective space to determine H(J). All of the results are proved for non-archimedean as well as archimedean absolute values. In what follows, when we discuss Philippon’s results, we shall consider only the archimedean case, and shall assume that the ring of coefficients is Z. We shall use the notation introduced above. At present there are three approaches to estimating the transcendence degree of number fields that are different in appearance, but are all based on the ideas described above. The first uses Philippon’s algebraic independence criterion (see [Philippon 1984, 1985, 1986]) and resembles Gel’fond’s proof of Theorems 6.1-6.3. The second approach, which is similar to the proof of Theorem 6.4, is based on direct estimates for ID(Z)] (see [Nesterenko 198413, 1985a, 19891). The third method makes use of a multidimensional generalization of Lemma 6.1 that is due to Brownawell [1987a, 19881. We shall examine all of these approaches, focusing our attention on how they obtain bounds for the transcendence degrees of the fields Li. 3.2. Philippon’s Criterion. In [1984, 19861 Philippon published a general theorem that included all of the known algebraic independence criteria as special cases and led to substantially improved bounds on transcendence degrees. Theorem 6.11. Let Ts = (WI,. . . , w,) E Cm, and let 6 be the prime ideal OfZ[Xl,... , x~] consisting of polynomials that vanish at GT. Let m - k, where 0 5 k 5 m, denote the codimension of this ideal. Suppose that CT,6, R, and S are increasing functions defined on N that for N E N suficiently large take values 1 1. Further suppose that CT+ 6 approaches +co as N + +co; the function S

(u + 6)6” is an increasing S(N)k+2

function;

-> c(a(N

and for all N E N suficiently

+ 1) + 6(N + l))S(N

+ 1)” (S(N)“+l

large one has + R(N

+ l)“+‘)

,

where c 2 1 is a constant depending only on m and 6. Under these conditions there does not exist a sequence of ideals 3~7 C Z[xl, . . . , x,] that have a

$3. Applications

of General

Elimination

Theory

finite number of zeros in the ball of radius exp(-R(N)) generated by polynomials QiN’, . . . , Q$$ for which degQiN) 2 5 6(N) >

279

centered at 15 and are

’ 1nH ~Q~‘)(LT), QtN’ 5 o(N), exp(-,S’(ii.l’*” ( >

0 < l<~txN) --

,m(N) ,

<

(1%

To obtain bounds on transcendence degrees, this theorem can be applied as follows. If we suppose that the number of algebraically independent numbers among wi , . . . , wm is small, i.e., that k is small, we can then use an analytic construction to obtain a sequence of ideals that satisfy all of the conditions in the theorem. This gives a contradiction, from which it follows that the transcendence degree of Q(wi, . . . , w,) must be large. We shall only give one corollary of this theorem (see [Philippon 1986]). Corollary 6.2. Suppose that i3 = (WI,. . . ,w,) E Cc”, n > m + 1, and cl > 0 is a constant that is suficiently large compared to m. There does not exist a sequence of homogeneous ideals 3~ = (P~,N, . . . , Pm(~),~) C Z[xl, . . . , xm], N 2 No, that have a finite number of zeros in the ball of radius exp(-3ciNq) ‘centered at ij and satisfy the inequalities t&N)

0

<

max l
< N, Ifi,N(ul,.

i=l . . ,%)I

, . . . , m(N) , 5 exp

(--clW)

.

In this case one takes ($7= (0), 6(N) = g(N) = N, R(N) = 3ciN, and = cl(N + l)“, where cl has been chosen to be sufficiently large compared to ~0. In applications of this and other corollaries of Theorem 6.11, the numbers wl, . . . , w, usually form a transcendence basis for the field under consideration. If we reformulate Corollary 6.2 without assuming that 772 m + 1, then if there exists a sequenceof ideals with the indicated properties, we can conclude that m > q - 1. This gives a lower bound for the transcendence degree m of the field generated by the wi. Using his algebraic independence criterion, Philippon was able to substantially improve the bounds on the transcendence degrees of the fields Li (see 11.1).

S(N)

Theorem 6.12. Let al,. . . , or, and bl, . . . , b, be complex numbers such that the following inequalities hold for any E > 0 and X 1 X(E), and for all vectors iG= (IQ,.. . , I+) E Zp and I= (ll,. . . ,l,) E Zq with Ihi 5 X and lliI 5 X:

lklal + . *. Then:

+ kpapl1 exp(--XE),

lllbl + . . . + l,b,l 2 exp(-X’)

. (20)

280

Chapter 6. Values of Functions That Have an Addition Law

1)

tr degQLl

> 5

- 1;

2) 3)

tr degqLz tr deg&3

2 s 2 5.

- 1;

In [1986] Philippon gives a brief proof of this theorem (using the example of part 3) to show how it follows from his criterion). He uses the analytic construction with “false variables” that was described in [Philippon 19811. In [1984a, 19861 Waldschmidt uses Philippon’s criterion to obtain a large number of results on the transcendence degree of fields generated by values of the exponential function and elliptic functions. In [1984a] Waldschmidt proves Theorem 6.12 as a consequence of a general theorem on algebraic groups. Another proof of Theorem 6.12, which instead of Philippon’s criterion uses a direct inductive bound on the ideals, is described in [Nesterenko 1985a], where part 2) is treated in detail (the method of proof will be discussed in $3.3 below). Finally, in [1987a] Brownawell proposes a third way to prove Theorem 6.12. Like Philippon [1981], he uses a construction with “false variables.” Brownawell investigates both the exponential and elliptic functions, and gives a detailed proof of the elliptic analogue of part 1) of Theorem 6.12. See $3.4 for more details on Brownawell’s work. In [1987] Diaz published a sketch of a proof of a further strengthening of the first inequality in Theorem 6.12, and he announced a result that strengthened the second inequality as well. In [Diaz 19891 he gave detailed proofs and also improved the technical conditions (20). We now state these theorems, which at present give the best bounds known for the transcendence degrees of LI and Lz. Theorem

4

6.13.

Let al,. . . , ap and bl, . . . , b, be complex numbers for which

there exists a constant X (bl , . . . , b,) > 0 such that the-following inequality holds for any X 2 X(bl, . . . , b,) and for all vectors 1 = (ll,. . . ,I,) E Zq with llil 5 X: ; lllbl +. . . + l,b,l 2 exp -Xpq/(2P+q) ( >

b)

there exists a constant X(al, . . . , aP) > 0 such that the following inequality holds for any X > X(al, . . . , ap) and for all vectors x = (ICI,. . . , kp) E Zp with llcil 2 x: Ihal + ... + kpapl 1 exp (-min

(XlnX,XPq/(P+2q)))

.

Then the following bound holds for pq > p-k q: tr degq LI > Theorem

6.14.

[ 1 P9 P+9

.

Let al,. . . , ap and bl , . . . , b, be complex numbers for which

$3. Applications

a)

of General Elimination

Theory

there exists a constant X(bl, . . . , b,) > 0 such that the following inequality holds for any X 2 X(bl,. . . , b,) and for all vectors i = (11,. . . , Zq)E I@ with lZiI 5 X: lllbl + ... + lqbql L exp -xP(Q+W(2P+d (

b)

281

; >

there exists a constant X(al, . . . , ap) > 0 such that the following inequality holds for any X 2 X(al,. . . , up) and for all vectors z = (ICI,. . . , kp) E ZP with lkil 5 X: lklal +*..+kpapl

>exp (-min(XlnX,Xp(q+‘)/(p+2q+1)))

.

Then the following bound holds for pq + p > 2: tr degQ L2 2

[ 1 Pq+P P+9

.

In particular, Theorem 6.14 implies a bound for the transcendence degree in Gel’fond’s conjecture that is the best that is currently known. Corollary 6.3. Let cr # 0,l be an algebraic number, and let /3 be an algebraic number of degree d > 2. Then tr deg&

(aP, CY~‘,. . . , oOOd-’)+gq.

To derive the corollary one takes p = q = d, ai = pi-‘, and bi = pi-l In (Y, i = l,..., d, in Theorem 6.14. In [1991] Ably proved an analogue of Corollary 6.3 in which Q! E A is replaced by a transcendental number that in some sensehas “good approximations” by rational numbers (see also [Amou 1991a]). Namely, he proved: If y is a transcendental number that has ‘good approximations” by rational numbers, and if /3 is an algebraic number of degree d 1 2, then tr deg&

(y, yO, yB2,. . . , yfidml )qg].

The improvement of the bound in Theorem 6.12 was achieved by changing the analytic construction of the polynomials needed in Philippon’s criterion. Diaz applied an idea of Chudnovsky (see p. 60-61 of [Chudnovsky 1982a]) in this situation. The result was that the auxiliary analytic functions were now functions of one variable, and their coefficients were polynomials in the numbers being studied, as they had been in Gel’fond’s original papers. We give a brief description of Diaz’s construction as it applies to Theorem 6.13. We set m = pq and w = (ealbl,. . . , e@q) E Cc”. We further set y = . , Ypq), and denote wll,..

282

Chapter

6. Values of Functions

That

Have an Addition

Law

Nd(L) = {(II, e.. 7Id) E Zd ; 0 5 Zi < Ly i = 1,. . . , d} * We let N denote a large natural number, and Xi, As,. . . denote positive constants that depend only on p, q, ai, and bj. In the first part of the proof Siegel’s lemma is used to construct a nontrivial set of polynomials

q EmT>

i = (ll,. ..,b> ENQ(L), L = ‘3’+lNp

7

such that the degree of 5 in each variable is less than 2P+‘NP+Q, the height Of Pi satisfies lnH(Pi) 5 Xi In N , and the following identity holds for all E = (Ici , . . . , lc,) E lVp(M) , M = NP: QI(y)

=

c e(v) SEW(L)

fr zf(y,j)ki” i=l

s 0.

(21)

j=l

For any vector r = (~11, . . . , rpq) E ;Zpq with rij 2 0 we denote

and Q#)

=

c iEN’

P&P)

fr

ti(Y.j)‘“i’j

i=l

j=l

.

(22)

We set R(N) = 8qNPq In N and let B be the ball of radius p = exp(-R(N)) centered at Z. We fix a point t E B and consider the set of all vectors 5; = (Tll, * . . , rpq) E Zpq, rij 2 0, for which there exists a polynomial Pi,?(Y) with PI?(~) # 0. We choose a vector in this set that has the smallest sum of its coordinates, and call it i?(t). The set of points f E B is infinite, but the set 93 = {F(if) ; f E B} is finite. We now choose MI = X2M with As E N sufficiently large, and consider the finite set of polynomials !3X = {Qz,JT)

; % E NP(M&

T E !R} .

(23)

It is to this set of polynomials that we apply Philippon’s criterion. It can be checked that deg Qx.,~ 5 AsNpfq,

In H(Qk,?) 5 AsNP+q,

Qz,, E ?JJZ7

and so one can take o(N) = 6(N) = XsNP+Q in Philippon’s criterion.

$3. Applications

To find an upper

of General

bound for IQ@)]

point $ = (
Elimination

Theory

283

we use the fact that there exists a

E B for which P = P(t), and we consider the auxiliary

At every point 77= kial + ... + ]cpup, E = (ki, . . . ,5)

E lV’(M),

we have

(24) For all z E Np(M) we have

c q,Jf)fr fi(Cp =0. idv’(L) i=lj=l

(25)

In fact, if we apply the operator D, to the identity (21), we obtain a sum of polynomials of the form &q(P) with polynomial coefficients that vanishes identically. We then use the fact that, by the definition of 7 = F(C), we have D+(E) = 0 for all operators 0~ of lower order than DF. We conclude that Q,,(c) = 0, which is the same as (25). Since the distance between z and t does not exceed the radius p = exp(-R(N)) of B, it follows from (24) and (25) that F(z) takes small values (of order p) at all of the points n = lciar +a . .+kpop, z = (ICI, . . . , Ic,) E Np(M). If we now apply the interpolation formula (see (29) in Chapter 3), we find that all of the values F(v)7

7 = klal + . . . + kpup,

x= (kl,..

. , kp) E NP(W)

at points of a much larger set are also small in absolute value. If we compare the representation (24) for F(q) and the representation (22) for Qh,+), we obtain the desired bound I&z&?)]

5 exp(-&NPQlnN),

K E NP(Ml),

FE R.

Thus, we can take S(N) = XJVP’J 1nN in Philippon’s criterion. It remains to verify the last condition, i.e., that the polynomials (23) have no common zeros in B. To do this we use Philippon’s theorem on the zeros of polynomials on algebraic groups (see [Philippon 19871). Sup_posethat all of the polynomials (23) vanished at a point c E B. Taking f = F(<), we find that

284

Chapter 6. Values of Functions That Have an Addition Law

for x E Np(Mi).

vanishes

This means that the polynomial

at all of the points

of the group (cC*)q. But this set of points is too large for this to be the case (for more details, see [Diaz 19891). Now let k be the maximum number of algebraically independent numbers in the set {ealbl ,...,eapbq}, and suppose that k + 1 5 pq/(p+ q). It is not hard to verify that all of the conditions of Philippon’s criterion are satisfied, and so we have obtained a contradiction. This means that k + 1 > pq/(p + q), i.e., tr deg Li 2 [pq/(p + q)]. The proof of Theorem 6.13 is complete. For other types of “algebraic independence criteria” that lead to quantitative results - i.e., bounds for the algebraic independence measure - see [Philippon 19851, [Jabbouri 19921, and [Ably 19921 (also see $5 below). 3.3. Direct Estimates for Ideals. The technique for proving algebraic independence that is described in this subsection is closely related to the method of proof of Theorem 6.4. However, instead of finding a lower bound for the value of a polynomial at a point, we shall make a similar estimate for the value of an ideal. This is how Nesterenko [1985a] proved Theorem 6.13, and also proved a theorem giving a bound for the algebraic independence measure of values at algebraic points of functions that satisfy certain functional equations (the Mahler equations). Following [Nesterenko 19891,we shall show how this approach can be used to prove part 2) of Theorem 6.13 (with somewhat more restrictive conditions on al,. . . , ap and bi, . . . , b,). We set m = pq + p, x = (Xos,Xij)i
woo = 1,

wio

= ai,

Wij

=

e a;bj

,

l
l<j
Theorem 6.15. Let n=P=&2, and let al,. . .,ap,bl,. . . , b, be complex numbers such that the following inequalities hold for some constant y > 0, for all X suficiently large, and for all nonzero vectors E = (ICI,. . . , kp) E Zp and ‘i = (11,. . . ,lq) E Zq with lkil 5 X and lljl 5 X: lklal +...

+ k,a, I 2 exp( -yX

lllbl + ... + l,b,l > exp(-yXlnX)

In X) , .

Then for any natural number r, 0 5 r 5 K - 1, there exists a constant pT = ,v(r, ai, bj) 2 0 such that the following inequality is satisfied for any unmixed homogeneous ideal J of the ring Z[x] for which h(3) = m + 1 - r:

$3. Applications

of General Elimination where

Theory

285

t(3) = deg3 + lnH(J)

.

From this theorem it follows that tr degQ LZ 1

[ 1 Pq+p Pf9

.

To seethis, we let T denote trdegq Lo, and let 6 denote the homogeneousprime ideal of Z[x] that is generated by all of the homogeneous polynomials that vanish at iZ. Then h(G) = m-r. If we apply the analogue of Lemma 6.1 to the ideal B and the polynomial PN E Z[F] in the sequence(15) that is constructed for L2 (using Gel’fond’s method), we obtain an unmixed homogeneous ideal 3N C Z[X] for which h(3N) = m - r + 1 and

If r satisfied the inequality r 5 K - 1, then (26) would contradict the lower bound for ]JN(z~) ] given by Theorem 6.15. Thus, r > K - 1, i.e., r 2 [K]. Theorem 6.15 is proved by induction on T. The inductive step is based on the ideas in 53.1. First of all, the theorem is reduced to the analogous statement for prime ideals - more precisely, to the following assertion: The set of real numbers S for which there exists a homogeneous prime ideal p satisfying the conditions h(p) = m + 1 - r,

Ip(

t(P) 5 8

< exp (-SK/(“-‘))

,

(27)

is bounded. To prove the above statement about prime ideals, for a given p satisfying (27) one constructs a corresponding homogeneous polynomial Q E Z[x] \ p that is sufficiently small at W. Applying the analogue of Lemma 6.1 to p and Q, we find an unmixed homogeneous ideal 3 c Z[Y] with h(3) = m - T+ 2 whose value at 15 is small compared to t(3). The resulting upper bound for ]a@)] contradicts the lower bound that one has from the induction assumption, and this concludes the inductive step. The construction of the polynomial Q uses the properties of p, and this is what distinguishes it from the construction in $3.2. The outward appearance of the construction is very close to that of Gel’fond’s original construction. The auxiliary analytic function is described in the following lemma (here the parameter T is given in some explicit way in terms of p and S). Lemma

6.5. Set

K = [X3Tqf1] ,

L

=

TP-‘(lnT)l/(Q+l) [

and

w = [TP+‘(lnT)-q/(q+l)] .

, I

286

Chapter 6. Values of Functions That Have an Addition

If S is suficiently large, then there exist homogeneous for-Z= (ICI,.. . , kP) E Zp, 0 5 ki < K, such that: 1) 2) 3)

Law

polynomials

Ax E Z[x]

t(AE) 5 XsP+Q(ln T)l/(q+l); not all of the Ax lie in the ideal p; the function of a complex variable F(z)

=

c

A~(gj)e(klal+...+k,ap)”

O
satisfies

the inequality l+)(11bl

for all nonnegative

+ ***+lqbq)l

I exp(-X4T”lnT)

integers 20, 11, . . . ,l, such that

osw<w,

O
(29)

In addition to Gel’fond’s classical argument, the proof of the lemma uses Chudnovsky’s idea of differentiating coefficients that was mentioned in 53.2. But instead of the set of points B(iZ, p) that was in Diaz’s construction, here one uses just a single point ZJ; and instead of the vanishing of polynomials one is interested in whether they lie in the ideal p. For details, see [Nesterenko 19891. This lemma is used in the proof in the following way. First, using the Hermite interpolation formula (see (29) in Chapter 3), one shows that IF(“)(lIbl + ... + l,b,)l is small on the somewhat larger set defined by the conditions 0 5 lj < XSL ) o<w<w, (30) where Xg >> 1, rather than by the conditions in (29) above. But the numbers F(W) (llbl+. . .+l,b,) are the values at z of certain homogeneous polynomials does not lie in p, then that Qwj(x) E Z[X]. If any of these polynomials polynomial can be chosen as Q, so that the analogue of Lemma 6.1 will give us a contradiction with the induction assumption, as explained above. Otherwise, we can use the fact that ]p(-)]w is very small (see (27)) to show that all of the polynomials Q, I(x) with w and i as in (30) also take very small values at Ts - much smaller, in fact, than the bound that comes from the Hermite interpolation formula. It turns out that the small size of these values, along with (30), is enough to prove that such a situation is possible only when all of the coefficients AK@) are very small. But then one can take the polynomial Q in our earlier argument to be any of the polynomials AL(X) that, by part 2) of the lemma, does not lie in p. The above method of argument is used in [Shestakova 19901 to prove a bound for the algebraic independence measure of ct? and c# when (Y is algebraic and p is a cubic irrationality; in [Shestakov 19921, [Nesterenko 19921, and [Nesterenko 19951 to prove bounds on the algebraic independence measure of

$3. Applications

of General

Elimination

Theory

287

values of elliptic functions; and in [Shestakov 19911to prove a bound on the transcendence degree of a field generated by values of elliptic functions. 3.4. Effective Hilbert Nullstellensatz. Let Pi, . . . , Pi be a set of polynomials in Z[zi , . . . ,z,] that have no common zeros. Then Hilbert’s Nullstellensatz says that there exist a nonzero integer a and polynomials A&) E Z[xl,... ,z,], 0 5 j 5 d, such that a= 5

Aj (z)Pj (:) ,

(31)

j=l

If we have upper bounds for deg Aj and H(Aj) (the maximum modulus of the coefficients of Aj) in terms of the degrees and size of the coefficients of the Pk,1I]CLd,thenforany~=((wl,...,w,)ECmwecanuse(31)tofinda lower bound for mal
rpm1

deg Pi 5 D,

I H,

l
Then there exist a E L&n<,a # 0, and polynomials Aj(Z) 1 5 j < d, such that the identity (31) holds and deg Ai 5 (8D)2M+‘,

lnl

, In] Ai

E %~[xl, . . . , xm],

1 < (8D)4M-’ (ln H + 80 ln(8D)) ,

where M = 2”-‘. Here and later, ] P(F) 1 denotes the maximum absolute value of all of the coefficients of P and their conjugates. Masser and Wiistholz [1983] actually prove and apply a somewhat more general result related to Hilbert Nullstellensatz. We have given the special case when the polynomials Pi have no common zeros. Theorem 6.16 implies the bound

lnl~~ydIPi@)l L -WI --

4M+1 (In H + 80 ln(8D)) - (8D)2M+1 In ]GJ], (32)

288

Chapter 6. Values of Functions That Have an Addition Law

where 1~1 = max( 1, ]wr 1,. . . , ]w~]). A series of papers ([Brownawell 1987b, 1987c], [Kollk 19881, [Philippon 1988a, 198813, 1989, 19911) subsequently appeared with improvements on the bounds in Theorem 6.16. We shall not dwell on these results, since they have not been used in proofs of algebraic independence. We now state a result of Philippon (Part I of [1983]) that he used to prove a lower bound for the transcendence degrees of certain fields generated by values of an elliptic function. Philippon needed a stronger hypothesis than the one in Theorem 6.16: the polynomials must have no common roots not only in affine space, but at infinity as well. If one takes homogeneous polynomials and assumes that they have no common zeros in projective space, it is possible to improve the bounds. Then some additional work is required in order to verify these stronger conditions if one wants to use results like Theorem 6.17 to prove that some numbers are algebraically independent. Theorem 6.17. Let IK be a finite extension of Q, and let Z = (wo, . . . , w,) be a nonzero set of complex numbers of which at most r + 1 are homogeneously algebraically independent over Q. There exists a constant cl > 0, depending only on m, wo, . . . , w,, and [IK : Q], such that for any polynomials &l(z),... , QT($ in &c[xo,. . . ,x,1 that have no common zeros in projective space and for any positive numbers 6 and o with

degQ@) L 6,

14&&N

I 0,

i=l,...,T,

one has the inequality lyJyT lQj@)l 2 exp (--cl(o + W’) --

.

(33)

The proof of Theorem 6.17 is based on an effective construction of the uresultants for a set of polynomials for which one has bounds on the coefficients. The bound on the right in (33) depends on r because the m - r polynomials that take part in the elimination of variables are the polynomial relations over Q. Their degrees and coefficients are, of course, among the we,...,w, bounded by constants that depend only on the wi. A result that is analogous to Theorem 6.17 was used in [Wiistholz 1983a]. A different proof of a theorem similar to Theorem 6.17 can be found in [Nesterenko 1985a]. If one constructs a set of polynomials that have no common zeros and take values at J that are sufficiently small compared to the degrees of the polynomials and the size of their coefficients, one can then use Theorems 6.16 and 6.17 to obtain lower bounds for the number of algebraically independent numbers in a given set of numbers. However, such a construction is not a simple matter. In [1987a, 19881Brownawell proved a result that he called a “local Liouville inequality.” It was a refinement of Theorem 6.17, with the conditions weakened in one respect. Instead of requiring that the polynomials have no common

$3. Applications

of General Elimination

Theory

289

zeros anywhere, one assumes only that they have no common zeros in a neighborhood of TS;the radius of this neighborhood appears in the bound. Like the results in Fjfj3.2-3.3, this inequality enables one to prove bounds on the transcendence degrees of the fields Li and their elliptic analogues (see [Brownawell 1987a] and [Brownawell and Tubbs 19871). We shall briefly illustrate this using the example of tr deg Ls in Theorem 6.14. We shall give Brownawell’s theorem in the form in [Brownawell1988] ( see Theorem B). We use the same notation as in $3.1, and set ]]ij]] = max,+~m ]wi] for ZJ = (WC,,. . . ,wm) E (Cm+‘. Theorem

ofiqxo,...

6.18. Let z E (Cm+’ \ {0}, let J be an unmixed homogeneousideal , xm], and let &I,. . . , Q,, be homogeneouspolynomials with

deg Qi 5 D,

H(Qi)

i = l,...,n,

I H,

that have no zeros in common with J in the ball of radius p centered at ~3, whereO
l&l@)1 13@)17

@jlldegQ1

IQn@)I ”

”

’ II~lldegQ,

2 -cl (Dj‘(DdegJ+

1

DlnH(J)

+lnHdeg3)

-Dump)

,

where ,u = min(n - 1, m + 1 - h(3)),

v = min(n,m + 1 - h(3)) ,

and cl > 0 is a constant that depends only on m. Under the conditions of Theorem 6.16 with lK = Q, one can obtain the following inequality, which is stronger than (32):

ln lyyd lpi(c)12 -Om -And under the conditions of Theorem 6.17, with 3 taken to be the ideal generated by all of the homogeneous polynomials in Z[xs, . . . , x,] that vanish at 15= (wc,... , w,), we obtain a result that generalizes Theorem 6.17 in the case lK = UJ. Namely, we have an inequality of the form

,~jyTIQ~@I L w(-cl(~+~-lnp)O --

.

Here the condition that the polynomials Qi have no common zeros in projective space is replaced by the condition that they have no common zeros in the ball of radius p centered at ij. In [Brownawell 19891 examples are given that show that the bounds in Theorem 6.18 are best possible in D, H, and p. In [1988] Brownawell remarks that the result can be carried over to polynomial rings over ZK, where lK is a finite extension of Q. Theorem 6.18 can be proved by induction on r = m + 1 - h(3) using the ideas in 53.1. One first reduces the bound to the case when J = p is a prime

290

Chapter 6. Values of Functions That Have an Addition

Law

ideal. If this ideal has no zeros in the same ball B that is in the hypothesis of the theorem, then the required bound follows immediately from the inequality in 53.1 that relates ]p@)] to the distance from z to the variety of zeros of p. On the other hand, if p has zeros in B, then, by assumption, there must be a polynomial Qi not lying in p. If we apply the analogue of Lemma 6.1 to this polynomial and the ideal p, and use the bound on la(Z)] in the induction assumption, we obtain the required inequality. In [1988] Brownawell used Theorem 6.18 to prove the following result. Theorem 6.19. Suppose that al,. . . ,ap,bl, . . . , b, E C have the property that for any E, 0 < E < l/24, there exists an injinite sequence of natural numbers N such that the following inequalities hold for all x = (ICI,. . . , Ic,) E Z* with lkil < N and all i = (II,. . . , lq) E Zq with lliI 5 N: lklal

+ ...

+ kpapl2 em C--W ,

lllbl + . . . + l,b,( 2 exp (-NE) trdego

Pq Ls 2 p+q

.

.

This theorem gives, exactly the same bound on the transcendence degree of L3 as Theorem 6.12. The difference is that the conditions in (35) are assumed to hold not for all sufficiently large N, but only for an infinite sequence of N. We have a similar situation with the other two parts of Theorem 6.12. The proof of Theorem 6.19 makes use of (34), which holds under the assumption that the polynomials Pi,. . . , Pd have no common zeros in the ball B(rj; p). The construction of the polynomials uses analytic functions in several auxiliary variables, just as in Philippon’s proof of Theorem 6.12. To prove that the resulting polynomials have no common zeros in B (z; exp (-4NE)), one uses the bound on the number of zeros of a polynomial on an algebraic group that Philippon proved in [1986b] and that we already referred to in 53.2. Finally, we note that the general argument in this subsection can be applied to find bounds for the transcendence degrees of all of the Li and the analogous fields generated by the values of elliptic functions. Brownawell [1987a] gives a set of parameters for all six cases, along with a detailed proof in the case of a field generated by the numbers p(aibj).

54. Algebraic Independence of the Values of Elliptic Functions We recall some notation that was introduced in connection with elliptic functions in 53 of Chapter 3. Let p(z) be the Weierstrass elliptic function with invariants gs and gs. Let k denote the field Q if g(z) does not have

54. Algebraic Independence of the Values of Elliptic Functions

291

complex multiplication; otherwise, k denotes the complex multiplication field. Let R denote the period lattice of a(z), with fundamental periods 2wi and 2~s. Let ~1 and 772 denote the quasiperiods corresponding to 2wi and 2wz (see 53.1 of Chapter 3). In the complex multiplication case r = w2/w1 is a quadratic imaginary irrationality. The function P(Z) satisfies an algebraic differential equation with constant coefficients, and has an addition law. These properties make ~(2) similar to the exponential function and enable one to prove theorems about its values that are analogous to those in the last section. This resemblance between p(z) and eZ is particularly close if gz and gs are assumed to be algebraic (in which case the differential equation satisfied by p(z) has algebraic coefficients) and if p(z) has complex multiplication. However, there are also some features of p(z) that make our work with elliptic functions more difficult. In the first place, a(z) is not an entire function, but only a meromorphic function. In addition, it has order of growth equal to 2 (rather than 1). On the other hand, it has the advantage of being doubly periodic. But the differential equation satisfied by p(z) is nonlinear, and the degrees of the polynomials relating the j-th derivative of p(z) with g(z) and p’(z) increase with j. The most important difficulty is the absence of theorems that bound the number of zeros of a polynomial in a set of functions that includes elliptic functions. A major improvement of the results in this area came about after ideas of commutative algebra were applied to estimate the multiplicity of zeros (see [Masser 19811, [Masser and Wiistholz 1981, 1983, 1985, 19861, [Philippon 1986b], [Wiistholz 1989a]). The ideas described in $3 of this chapter have also played an essential role. 4.1. Small Bounds for the Transcendence Degree. It follows from Schneider’s theorems [1937] ( see Theorems 3.8, 3.12, and 3.15) that if g2 and gs are algebraic, then 1) 2)

p(a) is transcendental for (Y E A, (Y # 0; g(Q) is transcendental if u E C, p(u) E A, and /3 E A, where p $! k and UP 4 0.

Clearly, these two results are elliptic analogues of Lindemann’s Theorem and Hilbert’s seventh problem. The first algebraic independence results for values of elliptic functions were proved in the 1970s by Shmelev, Brownawell, Kubota, and Chudnovsky. Shmelev [1976a] proved the following theorem giving two algebraically independent numbers in a set containing the values of both e” and p(z). Theorem

6.20. Suppose that ~1,. . . ,zs E @ are linearly independent over and none of the zi is a pole of p(/.~gz). Then there are {O), at least two numbers that are algebraically independent over Q among the 26 numbers i=1,...,8. g2, g3, zi, f+*', 64~24, (36)

Q,

PlYP2

E cc\

292

Chapter 6. Values of Functions That Have an Addition

Notice that in this theorem

Law

g2 and 93 are not assumed to be algebraic.

The next theorem is from [Shmelev 19811. Theorem 6.21. Suppose that ~1, ~2, p3 E C:, p3 # 0, and the following inequality holds for all ICI, k2 E Z with k = 1k1I + 1k2 1> 0:

lhcll + kw2l > exp

(-ykEi3(ln(k

+ l))“/‘)

,

where y > 0 does not depend on kl, k2. If ~1,. . . , ~10 E @ are linearly independent over Q, and if pszi 4 R for 1 5 i 5 10, then there are at least two numbers that are algebraically independent over Q among the 32 numbers 92,

93,

PzZ’,

epzzi,

dcl3-4,

i = l,...,lO.

These results are proved using Gel’fond’s method (see 31). The bounds for the number of zeros of the auxiliary function (which is a polynomial in the epjz and g(pgz)) reduce to the corresponding result for the exponential function. In [1973b] Waldschmidt published a general theorem giving conditions for the values of a given set of meromorphic functions to include a number that is transcendental over a field of finite transcendence type (see $1.4). Its proof is a generalization (in the spirit of the proof of Theorem 6.7) to fields of finite transcendence type of Schneider’s theorems on the values of meromorphic functions (see [Schneider 19571). In [1977] Brownawell and Kubota applied Waldschmidt’s theorem to sets of functions that include p(z), C(z), and e”. Here we shall give only two consequencesof the theorems in [Brownawell and Kubota 19771. Theorem 6.22. Let p(z) be a Weierstrass elliptic function with algebraic invariants g2 and 93 and with complex multiplication.

1) If CI:is an algebraic number of degree 4, [k(a) : k] = 2, and u E Cc,u # 0, then at least one of the numbers P(U),

@(U(y)> dua2),

dua3)

is defined and transcendental over the field Q(X). Thus, there are at least two numbers that are algebraically independent over Q among the five numbers r, P(U)> dua>7 P(u~2)Y PbQ3) . 2). If w is a period of p(z) and p is a cubic irrationality, the three numbers w, 6364, dwP2)

then at least two of

are algebraically independent over Q. These results are proved using theorems of Fel’dman that say that the transcendence type of Q(R) is at most 2 + E, and the transcendence type of Q(w) is at most 5 + E for any E > 0.

54. Algebraic Independence of the Values of Elliptic Functions

293

An approach based on the transcendence type of fields was also used in [Toyoda and Yasuda 19861. Among the consequences of the basic theorem they proved is the following result: if 2wi and 2wz are fundamental periods for an elliptic function with algebraic invariants and complex multiplication, then at least two of the numbers wl,

ln(2wi),

(2~1)~~‘~~

are algebraically independent over Q. In addition, using explicit formulas for the invariants of an elliptic function in terms of theta-functions, they prove the following result: For any z E Cc,Im(a) > 0, there are at least two numbers that are algebraically independent over Q among the five numbers M T?

z, q = eikr,

C

qnZ,

2

T&=--CO

(-l)“q”*

.

*=-CO

The next theorem appeared in [Chudnovsky 19761. Theorem 6.23. There are at least two numbers that are algebraically independent over Q among the six numbers Q2,

Q3,

(4,

WI,

59,

772 *

(37)

The proof of this theorem, which can be found in [Waldschmidt 1977a], is based on Gel’fond’s method. If Theorem 6.23 were. false, then, using the ‘Legendre relation ~2171 - ~1772 = xi , (38) we would see that all of the numbers in (37) are algebraic over Q(r). An auxiliary function is constructed as a polynomial in z, p(z) and C(z) with coefficients in Q[n]. The last step of the proof usesthe fact that the transcendence type of Q(n) is at most 2 + s. If we further assume that the elliptic function has algebraic invariants and complex multiplication, then we have the following result. Corollary 6.4. Suppose that p(z) has algebraic invariants and complex multiplication, and w is a nonzero period of a(z). Then K and w are algebraically independent. To prove this it suffices to observe that, under the conditions in the corollary, in addition to the relation (38) and the equality ws = rwi, we have another linear relation over A connecting the numbers ~1, r/s, w2 (see, for example, Lemma 3.1 of [Masser 19751). Here are two beautiful examples of this corollary. Consider the elliptic curves with complex multiplication given by the equations y2 = 4x3 - 4x and

294

Chapter 6. Values of Functions That Have an Addition Law y2 = 4x3 - 4 .

The corresponding w=2

elliptic functions

00

have periods

J1

and

Corollary 6.4 then says that x and r(1/4) are algebraically independent over Q, as are ?r and r(1/3). In particular, r(1/4) and r(1/3) are transcendental numbers. In [1996a, 1996b] Nesterenko proves a general theorem on algebraic independence of the values of the modular function (see $3.2 of Chapter 3) and its derivatives. Under the conditions of Corollary 6.4 this theorem implies that the numbers r, w and e2rrir (where Q(r) is the complex multiplication field) are algebraically independent. In particular, the numbers 7rIT,ea and r(1/4) are algebraically independent, as are the numbers X, eXfi and r(1/3). We also mention Wiistholz’s paper [1980], in which he proves a general theorem about the set of points at which three algebraically independent meromorphic functions take values in a fixed field of transcendence degree 1 (here the three functions satisfy algebraic differential equations). This theorem carries over Schneider’s theorem (see Theorem 3.17) to fields of transcendence degree 1. The proof is based on Schneider’s method, but instead of using Liouville’s theorem to get a lower bound for the nonzero values at algebraic points of polynomials with integer coefficients, Wiistholz uses Gel’fond’s algebraic independence criterion. The concrete consequences of this theorem for the values of an elliptic function are not as strong as Theorems 6.20 and 6.22. Finally, we take note of two papers by Tubbs [1987b, 199Oa], which are based on Gel’fond’s criterion and the bound in [Philippon 19871 for the multiplicity of zeros of polynomials on an algebraic group. Tubbs proves several general theorems on the existence of at least two algebraically independent numbers among a set of numbers that are connected with certain algebraic groups and analytic subgroups. The papers [Tubbs 1987b, 1990a] also contain many concrete consequences of the general results. Here are a few of them. a)

If (~1, (~2, a3, a4 are algebraic numbers and In (pi are values that are linearly independent over Q, then there are at least two algebraically independent numbers in the set Inal,

b)

lncr2,

lncq,

lncq,

dln4,

dlna2),

dlna3),

dlna4).

If ~1,212, u3 E C are linearly independent over Q, and if a(ui) E A, i = 1,2,3, then for any ,Cl# 0 there are at least two algebraically independent numbers in the set p,

ul,

212,

‘113,

epul,

epuz,

eou3

.

$4. Algebraic Independence of the Values of Elliptic Functions c)

295

If ul, ug E Cc and /31, /32 E C are two pairs of numbers that are each linearly independent over Q, and if p(uj) E A, j = 1,2, then there are at least two algebraically independent numbers in the set

4.2. Elliptic Analogues of the Lindemann -Weierstrass Theorem. In [198t)a] Chudnovsky proved the following result, which generalizes Schneider’s theorem. Theorem 6.24. Suppose that the Weierstrass function g(z) has algebraic invariants, and ~1,122, ~23 E A are linearly independent over Q. Then at least two of the numbers dd7

are algebraically

independent

dQ2),

64~3)

over Q.

In particular, this implies the following corollary, the Lindemann-Weierstrass theorem for two points.

which

is an analogue of

Corollary 6.5. Suppose that the Weierstrass function g(z) has algebraic invariants and complex multiplication, and ~1, cxz E A are linearly independent over k. Then the numbers da1

are algebraically

independent

1

and

P(~2)

over Q.

To prove this corollary it suffices to apply Theorem 6.24 to the numbers al, ~2, and crs = roi , where r = ws/wi is a quadratic imaginary irrationality that generates the field k. The proof of Theorem 6.24 is based on the method in 31. The auxiliary function is constructed in the form of a polynomial in z and p(z). Instead of Gel’fond’s algebraic independence criterion, one uses an analogue in which the degree and height of the polynomials are bounded separately (see [Brownawell 19751 and [Waldschmidt 1973a]). It was in [Chudnovsky 198Oa] that an important idea - one that makes it possible to neglect the increasing degree in g(z) of the successive derivatives of the auxiliary function - was first used. (This idea is sometimes called the Anderson-Baker-Coates-Masser technique.) Chudnovsky’s construction essentially opened up the possibility of studying the general case; however, the lack of any general algebraic independence criterion at that time (such a criterion was established only in 1986 by Philippon) stood in the way of obtaining a complete result. Incidentally, in [198Oa] Chudnovsky announced a result similar to Corollary 6.2 for any number n 5 6 ofpoints oi,...,on, and in [1982a] he even announced a stronger quantitative result for arbitrary n. In [198Oa] Chudnovsky also made the following conjecture.

296

Chapter

6. Values of Functions

That

Have an Addition

Law

Conjecture. Suppose that n 2 1, the Weierstrass function p(z) has algebraic invariants and doesnot have complex multiplication, and cyl, . . . , (Y, E A are linearly independent over Q. Then the numbers dm),.

. .7 P(4

(39)

are algebraically independent over UJ. This conjecture has only been proved for n = 1 (Schneider’s theorem). In [1982] Philippon announced a result (see Theorem 6.29 below) that in the special case ai = pi-‘, i = 1, . . . , n, with p an algebraic number of degree n, implies that among the numbers (39) there are at least n/4 that are algebraically independent over Q if g(z) has complex multiplication, and at least n/6 if p(z) does not have complex multiplication. In 1982 Wiistholz announced that the elliptic analogue of the Lindemann-Weierstrass theorem is true when g(z) has complex multiplication. Proofs of this theorem appeared in [Wiistholz 1983a] and Part II of [Philippon 19831. Theorem 6.25. Suppose that n > 1, the Weierstrass function p(z) has algebraic invariants and complex multiplication, and ~1,. . . , cr, E A are linearly independent over k. Then the numbers d4,

*. .,

dQ?d

are algebraically independent over Q. The papers [Wiistholz 1983a] and [Philippon 19831actually contain results about algebraic groups from which it follows (by applying them to the product of n elliptic curves) that if one does not assume complex multiplication, then among the numbers (39) there are at least n/2 that are algebraically independent over Q. If the same results are applied to the numbers CQ, XX~, i = l,... , n, where r is a generator of k over Q, then we obtain Theorem 6.25. The proofs of these general results in [Wiistholz 1983a] and [Philippon 19831 basically follow the plan outlined in 53.4. Using various theorems bounding the multiplicity of zeros of polynomials on algebraic groups and an auxiliary function with “false” variables, one constructs a set of polynomials that have no common zeros and take sufficiently small values at the point in question. The last stage of the proof consists in applying an effective Hilbert Nullstellensatz (see Theorem 17 in 53 of Part II of [Philippon 19831 and a similar result in [Wiistholz 1983a]). We note that subsequently a quantitative refinement of Theorem 6.25 (with a bound for the algebraic independence measure of the numbers) was proved (see $5). It should also be mentioned that [Nesterenko 19921contains a different proof of Theorem 6.25, based on the ideas in $3.3. 4.3. Elliptic Generalizations of Hilbert’s Seventh Problem. Suppose that +, E C are linearly independent over Q, and PI, . . . , ,OpE Ccare (unless otherwise stated) linearly independent over k. Let K1 denote the extension al,...,

$4. Algebraic field of Q that for which cr&

Independence

of the Values of Elliptic

Functions

297

is generated by all of the values p(cr;&), 1 5 i 5 p, 1 5 j 5 q, does not belong to the period lattice of P(Z). We further set

K4

=

K2

=Kl(%...,ap),

K3

=K1(LL...,Pq),

Kl(cq,...,

qdL...,PJ~

The second of the two results of Schneider at the beginning of $4.1 can be restated in the following equivalent form: 3) ifp > 1, then tr deg Kz 2 1. In [1982] Masser and Wiistholz announced a series of conditions under which the fields Ki have transcendence degree at least 2. Theorem 1) 2) 3) 4)

if if if if

pq pq pq pq

6.26. 2 > 2 >

Suppose

2p + 4q, then p + 4q, then 2p + 2q, then p + 2q, then

that Q(Z) has algebraic tr deg K1 tr deg K2 tr deg K3 tr deg KJ

invariants.

Then:

2 2; 2 2; 2 2; 2 2.

Detailed proofs of these elliptic analogues of Theorem 6.5 were published in [Masser und Wiistholz 19861. The same paper also contains the following analogue of Theorem 6.6 (see 51). invariants. In the case p = Theorem 6.27. Suppose that p(z) h as algebraic 2, q = 4, if CY~& 4 Q and p(alpj) E A for j = 1,2,3,4, then tr deg K4 2 2. In analogy to the theorem of Brownawell and Waldschmidt that says that at least one of the numbers eel eez must be transcendental, we have the following consequence of Theorem 6.27. Corollary 6.6. If p(z) h as algebraic then at least one of the numbers

is defined From theorem

invariants

and complex

multiplication,

and transcendental. part that

2) of Th eorem 6.26 one can obtain an elliptic Gel’fond proved in 1949 (see Theorem 6.3).

analogue

of the

Theorem 6.28. Suppose that the Weierstrass function p(z) has algebraic invariants and complex multiplication, (Y is a nonzero complex number not in the period lattice of p(z) f or which p(z) E A, and p is an algebraic number of and ~(a$~) are defined and algebraically degree 3. Then the numbers p(Q) independent over Q. The proofs of these results of Masser and Wiistholz make use of Gel’fond’s criterion (see Lemma 6.3) and some bounds on the multiplicities of the zeros at points of the form Icr,& +. . -+ k,/3, of polynomials in the functions p(ze+oiz),

298

Chapter 6. Values of Functions That Have an Addition Law

1 < i < p, with coefficients in cC[z], where the bounds depend on the degrees of the polynomials. These estimates, which replace Tijdeman’s theorem (see Lemma 6.4) in the elliptic case, are the main contribution of the paper [Masser and Wiistholz 19861. In [1982] and Part I of [1983], Philippon studied the special case when p = q = d, pj = p j-l, and oj = @j-l, where p is an algebraic number .of degree d 2 2 over Ic and u is an arbitrary complex number with UPS-’ $! R forj = l,..., d. Philippon proved the following theorem. Theorem 6.29. Under the above conditions, has algebraic invariants, then

if the Weierstrass function p(z)

In this situation, if g(u) E A and u7Ln R = {0}, then

These results are consequences of a general theorem on algebraic groups., We described the general plan of proof in 53.4. The algebraic part of the proof is based on a certain modification of Theorem 6.17 that generalizes Gel’fond’s criterion. The analytic construction of the auxiliary function uses “false” variables. An important role is played by the bound in [Philippon 19831 on the number of zeros of a polynomial on an algebraic group. If we take arbitrary cri and pi, then we need to assumethat certain arithmetic conditions hold. As in the exponential case, for the proofs to go through it is usually not enough just to have linear independence over Q or k. The required conditions - usually referred to as “technical conditions” - are that the following inequalities be fulfilled for some positive n: Itlcrl + . . . + t,a,l

> exp (-T”)

,

IQ4

2 exp C-W

,

+. . . + s&l

(40)

where

T = m4ltl I, . . . , ItpI>,

S = -4sl

I,. . . , hl> ,

for any ti ,..., t, and sr ,..., sq in the ring of multipliers of the lattice 0 and for all T and S sufficiently large. It has been conjectured that the results on transcendence degree that have been proved for both the exponential and elliptic functions remain true without any conditions of the type in (40). For some results in this direction, see [Delaunay 1992, 19931. In [1983] Masser and Wiistholz proved a logarithmic lower bound for the transcendence degree of Ki that is analogous to Theorem 6.10.

$4. Algebraic

Independence

of the Values of Elliptic

Functions

299

Theorem 6.30. Suppose that the Weierstrass function g(z) has algebraic invariants, and ~1,. . . , q, and 01,. . . , & are two sets of complex numbers that are each linearly independent over Q. Further supposethat the conditions (40) hold for some 6 2 0 (in this theorem one assumesthat sj E Z). Finally, assume that for some integer n > 1 one has Pcl P+%

> 2n+‘(n + 7) + 4n . -

Then the transcendence degree of Kl is at least n. Using bounds on the transcendence measures of e and 7rIT, we obtain the following simple consequence of Theorem 6.30. Corollary 6.7. For any natural number n and any integer N 2 3 . 2n+2(n + 7) , there are at least n numbers that are algebraically independent over Q among the numbers b”(e), de2L.. . , a(eN> , and also among the numbers P(n),

!h”>7...,

&w>.

The proof of Theorem 6.30 is based on the general argument in s3.4. The polynomials that are needed in order to apply Theorem 6.16 are constructed using a bound on the number of zeros of a polynomial on an algebraic group. In [1984a] Waldschmidt significantly improved and generalized Theorem 6.30. He was able to strengthen this theorem primarily because, instead of Theorem 6.16, he could use the algebraic independence criterion that Philippon had just proved (see Theorem 6.11). The basic result in [Waldschmidt 1984a] is a general theorem on algebraic groups. This theorem then leads to bounds on the transcendence degrees of fields generated by the values of the exponential and elliptic functions at certain points and by the points themselves. These results are analogous to Shmelev’s theorems discussed above, and they generalize the theorems in 53 on the values of the exponential function. Here we shall only give the special casesof these results that relate to the fields Ki, and we shall assume that the invariants gz and gs are algebraic (this is not assumed to be the case in [Waldschmidt 1984a], where gs and gs are included among the generators of the field). In order to derive the bounds Waldschmidt has to assume that the numbers (pi and pj satisfy (40) for some /C> 0. One then has Theorem

6.31.

Under the above conditions trdegKi

> - P9 P+%

- 1 ’

300

Chapter

6. Values of Functions

trdegKz

That

Have an Addition

Law

Pq+P 1 2 p+2q ’

trdegKs>‘s-I, trdegKq

2 -. Pq p+2q

In [1984a] Waldschmidt uses the bound on the number of zeros and the characterization of algebraic subgroups that are in [Masser and Wiistholz 19831. Waldschmidt’s paper [1984a] has a close relation with a later paper [Waldschmidt 19861, in which the same approach is used to study bounds on the transcendence degree, except that the estimates for the number of zeros in [Masser and Wiistholz 19831are replaced by those in [Philippon 19871.We shall give just one of the consequencesof the general theorem in [Waldschmidt 19861that concerns a set of distinct elliptic functions. Theorem 6.32. Let 01,. . . , C?dbe lattices in Ccfor which the corresponding elliptic curves El,. . . , Ed are not isogenous to one another, and let , @d(z) be the corresponding Weierstrass functions. Suppose that tab),... E C are linearly independent over Q and satisfy the following ~l,...,%n inequality for any & > 0, w E fli U ... U fld, and hl,...,h,,, E Z with , Ihnl) greater than some bound that depends on E: M=max(]hi],... Iw - hlvl - a.. - hmv,l

> exp (-M”)

.

Then there are at least md/(m + 2d) - 1 numbers that are algebraically independent over Q among the md numbers

Another proof of Theorem 6.31 was published in [Brownawell 1987a] and [Brownawell and Tubbs 19871.It used a different line of argument, of the type we described in $3.4. We note that the use of a “local Liouville inequality” (Theorem 6.18) instead of an algebraic independence criterion (Theorem 6.11) made it possible to somewhat weaken the technical conditions (40). Namely, it was enough for those conditions to hold for an infinite sequence of integers S rather than for all S sufficiently large (and similarly for T). Just as in the exponential case, in the elliptic case as well it was possible to strengthen the results by using Chudnovsky’s idea of differentiating the coefficients of the auxiliary function (see $3.2). In [1991] Shestakov proved the following theorem. Theorem 6.33. Suppose that the Weierstrass function p(z) has algebraic invariants gz and gs, and the complex numbers or,. . . , op and pi,. . . , & satisfy (40) with some positive constant K (K for the Qj is different from n for the pj; these values are given explicitly). Then:

54. Algebraic Independence of the Values of Elliptic Functions 1)

if pq/(p

2)

if (Pq+p)/(p+2q)

+ 2q) > 1, then

L 1, then

[ 1 -Pq+P p+2q

tr deg K2 2 3)

301

if (pq + 2q)/(p

;

+ 29) > 1, then tr deg KS 1

[ 1 P9 + 29 p+

If p(z) has complex multiplication and if the conditions in (40) hold for tj and sj in the ring of multipliers of R, then these bounds can be strengthened as follows: (41) Pq+P . P+9 We note that under these conditions a similar estimate for tr deg Ks follows from the bound for tr deg Kz. Prom Theorem 6.31 it is also possible to obtain ‘a bound for tr deg K4 under the same conditions. As a consequence of these results one can derive bounds that strengthen Theorem 6.29. trdegK2

2

[ 1

Theorem 6.34. Suppose that the Weierstrass function p(z) has algebraic invariants, /3 is an algebraic number of degree d over k, and u E C is such thatuPj$flforj=O,...,d-1. Then

[ I [ I

tr degQ(P(U),dub%.. . ,d@-‘)) > y

if

kz Q;

(42)

tr deg Q (P(U), duP), . . . , dd-‘))

if

k=Q.

(43)

2

%!$

In [1992] Ably published a proof of certain quantitative results that had the bounds (41)-(43) as consequences. He proved a generalization of Philippon’s criterion that enabled him to establish bounds on the algebraic independence measure. He then used these estimates to obtain his results by following the line of argument in $3.2. Shestakov’s proof, on the other hand, used the ideas in $3.3. We note that both papers [Shestakov 19911 and [Ably 19921 used the bounds on the number of zeros in [Philippon 1986133 and the results on algebraic subgroups in [Masser and Wiistholz 19831.

302

Chapter

6. Values

of F’unctions

$5. Quantitative

That

Have an Addition

Law

Results

Bounds on the Algebraic Independence Measure. Here for fixed , w,) E Cc” and arbitrary P E Z[si, . . . , x,] we want lower bounds on (WV... w,)] that depend on the degree and height of P. IP(w,..., The method that Gel’fond developed to prove bounds on the transcendence degree makes it possible not only to prove qualitative algebraic independence results, but also to obtain quantitative refinements, i.e., bounds on the algebraic independence measure. The first such result, a refinement of Theorem 6.3, was proved in [Gel’fond and Fel’dman 19501. 5.1.

Theorem 6.35. Suppose that (Y and 0 are algebraic numbers, where (Y # 0,l and p is a cubic irrationality. For any E > 0 there exists a positive constant c = c(a,P,e) such that the following inequality holds for any P E Z[x, y], P # 0: In IP (aD,c8’)l > - exp (c t(P)4+E) .

Recall that t(P) = deg P + In H(P). In 1974 Chudnovsky announced somegeneral results from which it followed, in particular, that the E in the exponent can be omitted in Theorem 6.35. However, no proof of these results was published. In [1979a] Brownawell proved that under the conditions in Theorem 6.35 one has lnIP(aP,aP2)I > - exp (c t(P)d$) , where dp = deg P. This inequality is a simple consequence of the following theorem in [Brownawell 1979a]. Theorem 6.36. Let Q be a nonzero complex number with In cr # 0, and let /3 be a cubic irrationality. There exists a positive constant c such that the following inequality holds for any relatively prime polynomials P(x, y), Q(x, y, z) E

ax,Y,4:

2 - exp(c d$d:(dp

In H(Q) + dQ In H(P) + dpd4))

.

(44)

Under the condition that CY,crp, and crpz are algebraically dependent, Theorem 6.36 implies a bound on the algebraic independence measure of any two of these numbers. In addition, if (Y is a number having good rational approximations, then it follows that (Y, c@, and & are in fact algebraically independent. The proof of this theorem is based on Gel’fond’s method, except that instead of Liouville’s theorem one uses an analogue of (44) in fewer variables - namely, the lower bound in [Mignotte and WaldSchmidt 19781 for lnmax(]A(a)], ]B( a, alo)]) for any relatively prime polynomials A(x), B(x, y) E Z[x, y].

55. Quantitative

Results

303

In [1987,1989] Diaz proved that the right side of the inequality in Theorem 6.35 can be replaced by - exp(c t(l’)2+E). In [1989] Diaz announced a bound on the algebraic independence measure of (~0 and af12 that is currently the best that we have. Theorem 6.37. Let cr be an algebraic number not equal to 0 or 1, and let p be a cubic irrationality. Then lnIP(aP,cuPZ)I for any noneero polynomial

> - exp(c t(P)dp)

P E Z[x, y].

A detailed proof of this theorem was published in [Diaz 19901. The proof used the method that was described in 53.2. A different proof was given in [Shestakov 19901, using the ideas in $3.3. In [1980a, 1982a] Chudnovsky studied bounds on the algebraic independence measure of numbers connected to elliptic functions. In [Chudnovsky 1982a] he gave a quantitative refinement of Corollary 6.4. Theorem 6.38. Suppose that the Weierstrass function g(z) has algebraic invariants and complex multiplication, w E R \ 20 is a period of g(z), and 77= 2C(w/2) is th e corresponding quasiperiod. Then ]R(~/w,~/w)]

2 exp (-c(lnH+dlnd)d2(lnd)2)

for any nonzero polynomial R E Z[x, y], where H = H(R), is a positive constant depending only on p(z).

(45) d = deg R, and c

The paper [Chudnovsky 1982a] was not written very clearly, and in places it contained false statements. However, there is no doubt that the ideas in it are of great interest and can be used to obtain a correct proof. The technique of differentiating the coefficients of the auxiliary function, which was later used by Diaz and others to prove bounds on the transcendence degree of fields (see §3), was one of the technical devices that appeared for the first time in this paper. Another was the use of the G-property (see 57 of Chapter 5) in the construction of the auxiliary function, which made it possible to omit the term In In H on the right side of (45). Chudnovsky’s attempts to use algebra-geometric methods to prove bounds on the transcendence degree have not been developed further - although there are analogies that can be drawn between the technical results in [Chudnovsky 1982a] on the dimension of 0 and the ideas that we discussed in $3. We note that Philibert [1988] and Jabbouri [1992], using the results in [Philippon 1986a], later proved inequalities that are weaker than (45); this was caused by peculiarities in their constructions of the auxiliary function. Prom Theorem 6.38 it follows that the transcendence type of the field a(w/n, v/ 7~) is at most 3 + E; it is this result that was proved in [Philibert 19881. Several other results were announced in [Chudnovsky 1982a], but without proofs.

304

Chapter 6. Values of Functions That Have an Addition Law

The first bound on the algebraic independence measure of the values of e” at algebraic points that are linearly independent over Q was proved by Mahler [1932] (see s4.3 of Chapter 2). Mahler’s inequality depended on parameters bounding the coefficients and degree of the polynomial in the estimate, and it held under the condition that the parameter H bounding the coefficients is larger than some Ho that depends not only on the points, but also on the degree of the polynomial. In [1977] Nesterenko gave explicit formulas for the parameters, not only for eL, but more generally for the broad class of Siegel E-functions (see $5.2 of Chapter 5). At the same time as attempts were being made to prove elliptic analogues of the Lindemann-Weierstrass theorem, some quantitative results were announced. In several papers (for example, [1980a, 1982a]) Chudnovsky announced bounds for the algebraic independence measure of &or), . . . , &(Y,) under the assumption that p(z) has algebraic invariants and complex multiplication, and the algebraic numbers ~1, . . . , (Y, are linearly independent over the complex multiplication field k. However, no proof has been published for m 2 2. After the appearance of [Wiistholz 1983b] and [Philippon 19831, where an elliptic analogue of the Lindemann-Weierstrass theorem was proved in the complex multiplication case (see Theorem 6.25), it also became possible to prove bounds on the algebraic independence measure. For a proof of the following theorem, see [Nesterenko 19921 and [Nesterenko 19951. Theorem 6.39. Suppose that the Weierstrass function g(z) has algebraic invariants and complex multiplication, m 2 1, and ~1, . . . , CY, E A are linearly independent over the complex multiplication field k. There exist positive constants cl and ~2, depending only on the numbers aj and the function g(z), such that, if H and d are any positive numbers with

In In H 2 cldm ln(d + 1) , and if P E Z[xl,... deg P 5 d, then

,x,1

(46)

is any nonzero polynomial with H(P)

IPM~I),

. . . , a(wn)>l 2 H-Q~”

.

5 H and (47)

We note that in the case m = 1 this result with a somewhat more stringent condition in place of (46) was proved in [Chudnovsky 198Oa]. The inequality (47) with m + E instead of m in the exponent on the right was first proved in [Jabbouri 19861as a consequence of a certain bound for abelian functions. In Jabbouri’s paper the right side of (46) had the form c3dm+E. Jabbouri used the approach in [Philippon 19831. The proof of Theorem 6.39 given in [Nesterenko 19921 and [Nesterenko 19951 is based on the ideas discussed in 53.3 of this chapter. 5.2. Bounds on Ideals, and the Algebraic Independence Measure. It is sometimes possible to prove quantitative results in cases when we are unable to prove algebraic independence of WI,. . . , w,, and even when we know

$5. Quantitative

Results

305

that these numbers are algebraically dependent over Q. For example, in the situation of Theorem 6.36 it is possible for the numbers (Y, c$, and cwfl*to be algebraically dependent - for instance, (Y might be an algebraic number and for (44) still to hold and in fact give us nontrivial information. In order to work with this situation, we must modify the notion of the algebraic independence measure. The next definition, which is taken from [Nesterenko 1983a], is based on concepts and notation that were introduced at the beginning of 53 of this chapter. Definition 6.4. Suppose that G = (~0,. . . ,wm) E (Cm+’ and r E N. The homogeneous algebraic independence measure of rank T of the numbers wo>***,wm is the function

@r(WO,.. . ,w,;T)

= @r(T) = min]J(Sj)] ,

where the minimum is taken over all unmixed ideals of rank m - r + 1 for which t(J) = H(J) + degJ 5 T. From the inequality (18) it follows that the function 9, (T) is closely related to the algebraic independence measure discussedin the previous subsection. If more than r of the numbers wc, . . . , w, are homogeneously algebraically independent, then the function @jr(T) takes positive values, and so it makes sense to speak of lower bounds for these values. It is sometimes useful to change the definition of G,. to make it a function of two variables H and d, where the condition t(3) < T is replaced by the conditions H(J) 5 H and deg3 5 d. The proofs of all of the results in 53.3 amount to successive estimates of the functions I, using induction on T in the appropriate range of values. An example is Theorem 6.15, which says that the following bound holds for the vector Z in the theorem and for all T such that T 5 K.- 1: @r(i3; T) > exp -prTTl(n-P)

,

where the p,. are constants that do not depend on T. The proof of Philippon’s algebraic independence criterion (see [Philippon 1986a] also essentially consists of a sequence of estimates of I, carried out under the conditions in the criterion. Moreover, in [1985] Philippon gives a version of Theorem 6.11 whose conclusion is a bound for @, for certain r. In [1992] Ably published a simpler and more convenient version of the algebraic independence criterion that made it possible to obtain quantitative results. In this paper Ably used his criterion to prove lower bounds for the functions cP,(ij; T) at points 55 that are the values of abelian functions (in particular, the values of exponential and elliptic functions). The paper [Jabbouri 19921contains a criterion that can be used to obtain bounds for @,.that depend on both H and d. For certain Lj and r, a lower bound for I,. enables one to obtain results similar to Theorem 6.34 for the numbers wc, . . . , wm. Such a result was first

306

Chapter 6. Values of J?unctions That Have an Addition Law

proved in [Philippon

19851 and [Nesterenko

1985a]. We shall give the version

in [Nesterenko1985aJ (seePropositions 5 and 6 of that paper). Theorem 1)

2)

6.40. Let 55 = (wo, . . . , w,)

E C?+‘,

m > 1.

If d + 1 is the maximum number of homogeneously algebraically independent numbers in the set (~0,. . . , wm}, then there exist positive constants Q, cl, and ~2, depending only on Z, such that

for T 2 Q and for any homogeneous polynomial P E Z[xo, . . . , x,] with P(Z) # 0 and t(P) 5 T. Suppose that wo = 1 and the set {WI,. . . , wd} contains a transcendence basis for the field Q(w1,. . . , w,). If RI,. . . , RN E Z[xo, . . . , xd] are homogeneous polynomials, and if the ideal (RI,. . . , RN) has rank d + 1 - r, then there exist positive constants cs, ~4, and ~5, depending only on Z, such that lyj”;i”N IRj(sS)l 2 Q,. (w; c4Td-‘+l) e-c6Td-‘+’ -if T > cg and t(Rj)

5 T, j = 1,. . . , N.

The next theorem is an example of the results that can be obtained Theorem 6.40 (see [Nesterenko 1985a]).

using

Theorem 6.41. Suppose that Q is an algebraic number not equal to 0 or 1, /3 is an algebraic number of degree d 2 2, and r is an integer satisfying 0 5 r < (d + 1)/2. Th en f or any v in the interval r < v < (d + 1)/2 there exists a positive constant p = P((Y, p, Y) such that the inequality ,...,cx

,,-,>

1 >

exp

(++-+/(“-1)

holds for any set of polynomials Rj E Z[xl, . . . ,x&l], j = 1,. . . , N, that generates an ideal of rank d - r in the ring Z[xl, . . . , x&I] and for any T _> t(Rj), j = 1,. . . , N. A similar theorem will the case d = of this result

theorem was also proved in [Philippon 19851. We note that this not give us a bound for the algebraic independence measure in 3 (Theorems 6.35 and 6.37). In [Ably 19921 there is a refinement - namely, the inequality o!, aP2 ,...,cY pdvl) 1 2 exp (- exp@T2))

under the same conditions 6.37 when r = 3.

with

r = (d + 1)/2 -

that agrees with

Theorem

5.3. The Approximation Measure. There is another approach to proving quantitative results in situations where we have not proved that the numbers

$5. Quantitative

Results

307

in question are algebraically independent. We shall use the notion of the size of a number that was introduced in $1.4. Recall that the size s(o) was defined for numbers a that lie in finitely generated extensions of Q. If it is known that more than d of the numbers wi , . . . , w,,, are algebraically independent over Q, then for any field lK > Q with tr dego K = d and for any m-tuple
is nonzero, and it makes sense to ask for lower bounds for this number that depend on maxi~j~m s(&). In the case d = 0 and lK = Q this is the classical problem of simultaneous approximation by rational numbers. If K is an algebraic extension of Q, then theorems on simultaneous approximation of the logarithms of algebraic numbers by algebraic numbers (see Theorem 3.34) are examples of this kind of result. The first result of this type for fields of positive transcendence degree was obtained by Shmelev [1972b]. Theorem 6.42. Let (~1 = 1, ~2, ~23 and /31,p2 be sets that satisfy the inequalities 1x1 + x2a2

+ 23~x31

IYIPI +

~2P2l

2 exp(-q&E), 2 ed--72y21ny),

x =

of complex numbers

Ial+

1x21+

1x31 ;

Y = IYll + IYZI >

for all integers xi with x > 0 and all integers yi with y > 0, and for some positive constants ~1 and ~2. If IK is a finite algebraic eztension of Q(n), then any set of 10 numbers COl,

Q2,

C207

C307

Cij

E K

l
l<j<2,

(48)

satisfies the inequality

Ia2

-

czol

+ Ia3

-601

+ IPl

-

~011 + I@2 -

(021 + +Ik i=l

j=l

/e"'flj

-CjI

2

2 exp ( -/M22(ln s)35) , where s is the maximum size of the numbers (48) and p is a constant that depends only on the ai and /3j. Fel’dman’s bound on the transcendence measure of 7r played an essential role in the proof of Theorem 6.42. In [1976a] Shmelev proved a similar result for arbitrary fields lK of transcendence degree 1 and finite transcendence type. The papers [Shmelev 1975b,1985] are devoted to the same theme, but consider approximations by elements of an arbitrary field lK of transcendence degree 1; the bounds on the transcendence type of the field are replaced by arguments using the resultant. In this context one introduces the following definition.

308

Chapter

6. Values of Functions

That

Have an Addition

Law

Definition 6.5. Suppose that W = (WI,. . . ,w,) E Cc” and K is a finitely generated extension of Q. The simultaneozls approximation measure of the numbers wr , . . . , w, by the field lK is the function EK (w1,...,wm;

T) = %(T)

= min rrnjym ]wj - <j ] , --

where the minimum is taken over all m-tuples 51,. . . , &,, E lK for which s(&) 5 T,j = l,..., m. In [1985] Philippon proved that for any d 1 1 there is a relation between Grn-d - i.e., the algebraic independence measure of rank m - d - and the approximation measure by fields of transcendence degree d. Theorem 6.43. Suppose that 15= (WC,,.. . , w,) E Cm+‘, wg = 1, and R is an extension of Q of transcendence degreed. There exist constants cl and cg, depending only on CJand IK, such that EK(Z,T)

> Qrn--d (clTm) e- C2Trn

.

In [1991] Ably published a series of results giving lower bounds for the approximation measure of a set of values of abelian functions (and, in particular,, exponential or elliptic functions) by fields of large transcendence degree. His method of proof was based on the ideas in s3.2.

References

309

References* Ably M. (1989): Mesure d’approximation simultanee. Ann. Fat. Sci. Univ. Toulouse, V. Ser., Math. 10, 259-289. Zbl 705.11035 Ably M. (1991): Independance algebrique de certains nombres admettant une “bonne” approximation diophantienne. C.R. Acad. Sci., Paris, Ser. A 313, 653655. Zbl 758.11033 Ably M. (1992): Resultats quantitatifs d’independance algebrique pour les groupes aigebriques. J. Number Theory 42, 194-231. Zbl 758.11032 Aleksandrov P.S., ed. (1971): Die Hilbertschen Probleme. Geest u. Portig, Leipzig. Zbl 242.00004 Alladi K., Robinson M. (1979): 0 n certain irrational values of the logarithm. Lect. Notes Math. 751, l-9. Zbl 414.10031 Alladi K., Robinson M. (1980): Legendre polynomials and irrationality. J. Reine Angew. Math. 318, 137-155. Zbl 425.10039 Amoroso F. (1990): Polynomials with high multiplicity. Acta Arith. 56, 345-364. Zbl 715.11034 Amoroso F. (1994): Values of polynomials with integer coefficients and distance to their common zeros, Acta Arith. 68, 101-112. Zbl 715.11034 Amou M. (1991a): Algebraic independence of the values of certain functions at a transcendental number. Acta Arith. 59, 71-82. Zbl 735.11031 Amou M. (1991b): Approximation to certain transcendental decimal fractions by algebraic numbers. J. Number Theory 37, 231-241. Zbl 718.11030 Anderson M. (1977): Inhomogeneous linear forms in algebraic points of an elliptic function. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, 121-143. Zbl 362.10030 Andre Y. (1989): G-Functions and Geometry. Friedr. Vieweg & Sohn, Braunschwieg/Wiesbaden. Zbl 688.10032 Anfert’eva, E.A., Chudakov N.G. (1968) : M inima of the norm function in imaginary quadratic fields. Dokl. Akad. Nauk SSSR, Ser. Mat. 183, 255-256. English transl.: Sov. Math., Dokl. 9, 1342-1344. Zbl 198.07302 Angelesko M.A. (1919): Sur deux extensions des fractions continues algebriques. C.R. Acad. Sci., Paris, Ser. A 168, 262-265. JFM 47.0431.02 Apery R. (1979): Irrationahte de C(2) et C(3). Asterisque 61, 11-13. Zbl 401.10049 Arnol’d V.I. (1983): Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, Berlin Heidelberg New York. Zbl 507.34003 Babaev G.B. (1966): Distribution of Integral Points on Algebraic Surfaces. Tadzhik. Gosudarstv. Univ., Dushanbe Baker A. (1964a): Approximations to the logarithms of certain rational numbers. Acta Arith. 10, 315-323. Zbl 201.37603 Baker A. (196413): Rational approximation to certain algebraic numbers. Proc. Lond. Math. Sot. 14, 385-398. Zbl 131.29102 Baker A. (1964c): Rational approximation to fi and other algebraic numbers. Q. J. Math., Oxf. II. Ser. 15, 375-383. Zbl 222.10036 Baker A. (1965): On some Diophantine inequalities involving the exponential function. Can. J. Math. 17, 616-626. Zbl 147.30901 * For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl), compiled by means of the MATH database, and Jahrbuch iiber die Fortschritte der Mathematik (JFM) have, as far as possible, been included in this bibliography.

310 Baker

References

A. (1966a): Linear forms in the logarithms of algebraic numbers. I. Mathe13, 204-216. Zbl 161.05201 Baker A. (1966b): A note on the Pade table. Indag. Math. t&596-601. Zbll99.09502 Baker A. (1967a): Simultaneous rational approximations to certain algebraic numbers. Proc. Cambridge Philos. Sot. 63, 693-702. Zbl 166.05503 Baker A. (196713): Linear forms in the logarithms of algebraic numbers. II. Mathematika 14, 102-107. Zbl 161.05202 Baker A. (1967c): Linear forms in the logarithms of algebraic numbers. III. Mathemat&a 14, 220-228. Zbl 161.05307 Baker A. (1968a): Linear forms in the logarithms of algebraic numbers. IV. Mathematika 15, 204-216. Zbl 169.37802 Baker A. (1968b): Contributions to the theory of Diophantine equations. Phi10s. Trans. R. Sot. Lond., Ser. A 263, 173-191. Zbl 157.09702 Baker A. (1968c): Contributions to the theory of Diophantine equations. II. Phi10s. Trans. R. Sot. Lond., Ser. A 263, 193-208. Zbl 157.09801 Baker A. (1968d): The Diophantine equation y2 = ax3+bx2+cx+d. J. Lond. Math. Sot. 43, l-9. Zbl 155.08701 Baker A. (1969a): Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Sot. 65, 439-444. Zbl 174.33803 Baker A. (1969b): A remark on the class number of quadratic fields. Bull. Lond. Math. Sot. 1, 98-102. Zbl 177.07403 Baker A. (1969c): On the quasi-periods of the Weierstrass C-function. Nachr. Akad. Wiss. Giitt., II. Math.-Phys. 16, 145-157. Zbl 201.05403 Baker A. (1970a): On the periods of the Weierstrass pfunction. Symp. Math. 4, 155-174. Zbl 223.10019 Baker A. (1970b): An estimate for the gfunction at an algebraic point. Am. J. Math. 92, 619-622. Zbl 208.30901 Baker A. (1971a): Imaginary quadratic fields with class number 2. Ann. Math., II. Ser. 94, 139-152. Zbl 219.12008 Baker A. (1971b): On the class number of imaginary quadratic fields. Bull. Am. Math. Sot. 77, 678-684. Zbl 221.12006 Baker A. (1972): A sharpening of the bounds for linear forms in logarithms. I. Acta Arith. 21, 117-129. Zbl 244.10031 Baker A. (1973a): A sharphening of the bounds for linear forms in logarithms. II. Acta Arith. 24, 33-36. Zbl 261.10025 Baker A. (1973b): A central theorem in transcendence theory. In: Diophantine Approximation and Its Applications. Academic Press, London New York, l-23. Zbl 262.10022 Baker A. (1973c): Recent advances in transcendence theory. Tr. Mat. Inst. Steklova 132, 67-69. Zbl 291.10027 Baker A. (1975a): Transcendental Number Theory. Cambridge Univ. Press, Cambridge. Zbl 297.10013 Baker A. (1975b): A sharpening of the bounds for linear forms in logarithms. III. Acta. Arith. 27, 247-252. Zbl 301.10030 Baker A. (1977): The theory of linear forms in logarithms. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, l-27. Zbl 361.10028 Baker A., Birch B.J., Wirsing E.A. (1973): 0 n a problem of Chowla. J. Number Theory 5, 224-236. Zbl 267.10065 Baker A., Coates J. (1970): Integer points on curves of genus 1. Proc. Cambridge Philos. Sot. 67, 595-602. Zbl 194.07601 Baker A., Coates J. (1975): Fractional parts of powers of rationals. Math. Proc. Cambridge Philos. Sot. 77, 269-279. Zbl 298.10018

math

References

311

Baker A., Schinzel A. (1971): On the least integers represented by the genera of binary quadratic forms. Acta Arith. 18, 137-144. Zbl 218.10040 Baker A., Stark H.M. (1971): On a fundamental inequality in number theory. Ann. Math., II. Ser. 94, 190-199. Zbl 219.12009 Baker A., Stewart C.L. (1988): On effective approximations to cubic irrationals. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, l-24. Zbl 656.10025 Baker A., Wiistholz G. (1993): Logarithmic forms and group varieties. J. Reine Angew. Math. 442, 19-62. Zbl 788.11026 Barre-Sirieix K., Diaz G., Gramain F., Philibert G. (1996): Une preuve de la conjecture de Mahler-Manin. Invent. Math. 124, l-9 Bassalygo L.A., Fel’dman N.I., Leont’ev V.K., Zinov’ev V.A. (1975): Nonexistence of perfect codes for certain composite alphabets. Probl. Peredaci Inf. 11, 3-13. Zbl 326.94008 Becker P.G. (1991a): Effective measures for algebraic independence of the values of Mahler type functions. Acta Arith. 58, 239-250. Zbl 735.11030 Becker P.G. (1991b): Exponential Diophantine equations and the irrationality of certain real numbers. J. Number Theory 39, 108-116. Zbl 733.11010 Becker-Landeck P.G. (1986): Transcendence measure by Mahler’s transcendence method. Bull. Aust. Math. Sot. 33, 59-65. Zbl 571.10033 Belogrivov 1.1. (1967): Transcendence and algebraic independence of values of hypergeometric E-functions. Dokl. Akad. Nauk. Zbl204.06804 SSSR, Ser. Mat. 174, 267-270. English transl.: Sov. Math., Dokl. 8, 610-613 Belogrivov 1.1. (1970): Transcendence and algebraic independence of the values of some hypergeometric E-functions. Mat. Sb. 82 (124), 387-408. English transl.: Math. USSR, Sb. 11, 355-376. Zbl 225.10036 Belogrivov 1.1. (1971): On transcendence and algebraic independence of the values of some hypergeometric E-functions. Sib. Mat. Zh. 12, 961-982. English transl.: Sib. Math. J. 12, 690-705. Zbl 229.10018 Bertrand D. (1975/1976): Transcendance de valeurs de la fonction gamma. Skmin. Delange-Pisot-Poitou, Theor. Nombres 17, 8/l-8/8. Zbl 352.10016 Bertrand D. (1977): A transcendence criterion for meromorphic functions. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, 187-193. Zbl 365.10024 Bertrand D. (1979): Fonctions modulaires et independance algebrique. II. Asterisque 61, 29-34. Zbl 414.10035 Bertrand D. (1979/1980): Varietes abeliennes et formes linkaires d’integrales elliptiques. Semin. Dklange-Pisot-Poitou, Theor. Nombres 21, 15-28. Zbl 452.10038 Bertrand D. (1980): Transcendental properties of abelian integrals. Queen’s Pap. Pure Appl. Math 54, 309-318. Zbl 452.10037 Bertrand D. (1982): Abelian functions and transcendence. Lond. Math. Sot. Lect. Note Ser. 56, l-10. Zbl 488.10030 Bertrand D. (1983): Endomorphismes de groupes algebriques; applications arithmetiques. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. .BirkhBuser, Base1 Boston, l-45. Zbl 526.10029 Bertrand D. (1984): Proprietes arithmetiques de fonctions theta a plusieurs variables. Lect. Notes Math. 1068, 17-22. Zbl 546.14029 Bertrand D. (1985): Exposants, irregularitks et multiplicites. Groupe Etude Anal. Ultrametrique, 1983/84, No. 10, l-6. Zbl 589.12019 Bertrand D. (1987): Lemmes de zeros et nombres transcendants. Asterisque 145-146, 21-44. Zbl 613.14001 Bertrand D. (1987/1988): Exposants des systemes differentiels, vecteurs cycliques et majorations de multiplicites. Publ. Math. Univ. Paris VI, No. 88, 1-17

312

References

Bertrand D. (1992): Un analogue differentiel de la theorie de Kummer. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 39-49. Zbl 785.12002 Bertrand D., Beukers F. (1985): Equations differentielles lineaires, et majorations de multiplicites. Ann. Sci. EC. Norm. Super., IV. Ser. 18, 181-192. Zbl 578.12016 Bertrand D., Masser D. (1980a): Linear forms in elliptic integrals. Invent. Math. 58, 283-288. Zbl 425.10041 Bertrand D., Masser D. (1980b): Formes lineaires d’integrales abeliennes. CR. Acad. Sci., Paris, Ser. A 290, 725-727. Zbl 442.10025 Beukers F. (1979): A note on the irrationality of C(2) and C(3). Bull. Lond. Math. Sot. 11, 268-272. Zbl 421.10023 Beukers F. (1980): Legendre polynomials in irrationality proofs. Bull. Aust. Math. Sot. 22, 431-438. Zbl 436.10016 Beukers F. (1981): Pade-approximations in number theory. Lect. Notes Math. 888, 90-99. Zbl 478.10016 Beukers F. (1983): Irrationality of 7r2 , periods of an elliptic curve and ri(5). In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhauser, Base1 Boston, 47-66. Zbl 518.10040 Beukers F. (1985): Some congruences for the Apery numbers. J. Number Theory 21, 141-155. Zbl 571.10008 Beukers F. (1988): Some new results on algebraic independence of E-functions. In: New Advances in Transcendence Theory. Zbl 656.10029 Cambridge Univ. Press, Cambridge, 56-67 Beukers F. (1991/93): Algebraic values of G-functions. J. Reine Angew. Math. 434, 45-65. Zbl 753.11024 Beukers F., Brownawell W.D., Heckman G. (1988): Siegel normality. Ann. Math., II. Ser. 127, 279-308. Zbl 652.10027 Beukers F., Bezivin J.P., Robba P. (1990): An alternative proof of the LindemannWeierstrass theorem. Am. Math. Mon. 97, 193-197. Zbl 742.11038 Beukers F., Matala-Aho T., V%?in%nen K. (1983): Remarks on the arithmetic properties of the values of hypergeometric functions. Acta Arith. 42, 281-289. Zbl 527.10032 Beukers F., Wolfart G. (1988): Algebraic values of hypergeometric functions. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 68-81. Zbl 656.10030 Bezivin J.P. (1988): Independance lineaire des valeurs des solutions transcendantes de certaines equations fonctionnelles. I. Manuscr. Math. 61, 103-129. Zbl 644.10025 Bezivin J.P. (1990): Independance lineaire des valeurs des solutions transcendantes de certaines equations fonctionnelles. II. Acta Arith. 55, 233-240. Zbl 712.11038 BCzivin J.P, Robba P. (1989): A new p-adic method for proving irrationality and transcendence results. Ann. Math., II. Ser. 129, 151-160. Zbl 679.10025 Bijlsma A. (1977): On the simultaneous approximation of a, b, and ab. Compos. Math. 35, 99-111. Zbl 355.10025 Bijlsma A. (1978): Simultaneous approximations in transcendental number theory. Math. Centre Tracts, No. 94, 101 p. Zbl 414.10029 Bijlsma A. (1980): An elliptic analogue of the Franklin-Schneider theorem. Ann. Fat. Sci. Toulouse, V. Ser., Math. 11, 101-116. Zbl 447.10034 Blass J., Glass A.M.W., Manski D.K., Meronk D.B., Steiner R.P. (1990): Culistants for lower bounds for linear forms in the logarithms of algebraic numbers. Acta Arith. 55, 1-14 and 15-22. Zbl 709.11037 Boehle K. (1933): Uber die Transzendenz von Potenzen mit algebraischen Exponenten. Math. Ann. 108, 56-74. Zbl 006.15701

References

313

Boehle K. (1935): ober die Approximation von Potenzen mit algebraischen Exponenten durch algebraische Zahlen. Math. Ann. 110, 662-678. Zbl 010.25101 Bombieri E. (1970): Algebraic values of meromorphic maps. Invent. Math. 10, 267287. Zbl 214.33702 Bombieri E. (1981): On G-functions. In: Recent Progress in Analytic Number Theory, Vol. 2. Academic Press, London New York, l-67. Zbl 461.10031 Bombieri E. (1982): On the Thins-Siegel-Dyson theorem. Acta Math. 148, 255-296. Zbl 505.10015 Bombieri E. (1985): On the Thue-Mahler equation. Lect. Notes Math. 1290, 213-243. Zbl 646.10009 Bombieri E. (1990): The Mordell conjecture revisited. Ann. SC. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 615-640. Zbl 722.14010 Bombieri E., Mueller J. (1983): On effective measures of irrationality for (a/b)‘/‘. J. Reine Angew. Math. 342, 173-196. Zbl 516.10024 Bombieri E., Schmidt W. (1987): On Thue’s equation. Invent. Math. 88, 69-81. Zbl 614.10018 Bombieri E., van der Poorten A. (1988): Some quantitative results related to Roth’s theorem. J. Aust. Math. Sot. 45, 233-248. Zbl 664.10017 Bore1 E. (1899): Sur la nature arithmetique du nombre e. C.R. Acad. Sci., Paris, Ser. A 128, 596-599. JFM 30.0384.03 Borevich Z.I., Shafarevich I.R. (1966): N umber Theory. Academic Press, London New York. Zbl 145.04902 Borwein P.B. (1991): On the irrationality of c(qn + r)-‘. J. Number Theory 37, 253-259. Zbl 718.11029 Brownawell W.D. (1972): Some transcendence results for the exponential function. Norske Vid. Selsk. Skr. 11, l-2. Zbl 234.12001 Brownawell W.D. (1974): The algebraic independence of certain numbers related by the exponential function. J. Number Theory 6, 22-31. Zbl 275.10020 Brownawell W.D. (1975): Gelfond’s method for algebraic independence. Trans. Am. Math. Sot. 210, l-26. Zbl 312.10022 Brownawell W.D. (1975/1976): Pairs of polynomials small at a number to certain algebraic powers. Semin. Ddlange-Pisot-Poitou, Theor. Nombres 17, ll/Ol-11/12. Zbl 351.10022 Brownawell W.D. (1979a): On the Gelfond-Feldman measure of algebraic independence. Compos. Math. 38, 355-368. Zbl 402.10039 Brownawell W.D. (1979b): On the development of Gelfond’s method. Lect. Notes Math. 751, 18-44. Zbl 418.10032 Brownawell W.D. (1980): On the orders of zero of certain functions. MCm. Sot. Math. Fr., Nouv. Ser. 108, No. 2, 5-20. Zbl 467.10024 Brownawell W.D. (1983): Zero estimates for solutions of differential equations. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhluser, Base1 Boston, 67-94. Zbl 513.10034 Brownawell W.D. (1985): Effectivity in independence measures for values of Efunctions. J. Aust. Math. Sot. 39, 227-240. Zbl 574.10039 Brownawell W.D. (1986a): An “ideal” transcendence measure. Lond. Math. Sot. Lect. Note Ser. 109, 1-19. Zbl 589.10033 Brownawell W.D. (1986b): The effectivity of certain measures of algebraic independence. Semin. Thkor. Nombres, Paris 1984/1985, Birkhiuser, Base1 Boston, 41-50. Zbl 603.10035 Brownawell W.D. (1987a): Large transcendence degree revisited. I. Exponential and non-CM cases. Lect. Notes Math. 1290, 149-173. Zbl 626.10030 Brownawell W.D. (1987b): Bounds for the degree in the Nullstellensatz. Ann. Math., II. Ser. 126, 577-591. Zbl 641.14001

314

References

Brownawell W.D. (1987c): Borne effective pour l’exposant dans le theoreme des zeros. C.R. Acad. Sci., Paris, Ser. A 305, 287-290. Zbl 648.13011 Brownawell W.D. (1988): Local Diophantine Nullstellen inequalities. J. Am. Math. Sot. 1, 311-322. Zbl 651.10022 Brownawell W.D. (1989): Applications of Cayley-Chow forms. Lect. Notes Math. 1380, 1-18. Zbl 674.10031 Brownawell W.D., Kubota K.K. (1977): The algebraic independence of Weierstrass functions and some related numbers. Acta Arith. 33, 111-149. Zbl 356.10027 Brownawell W.D., Masser D.W. (1980a): Multiplicity estimates for analytic functions. I. J. Reine Angew. Math. 314, 200-216. Zbl 417.10027 Brownawell W.D., Masser D.W. (1980b): Multiplicity estimates for analytic functions. II. Duke Math. J. 47, 273-295. Zbl 461.10027 Brownawell W.D., Tubbs R. (1987): Large transcendence degree revisited. II. The CM case. Lect. Notes Math. 1290, 175-188. Zbl 626.10031 Brownawell W.D., Waldschmidt M. (1977): The algebraic independence of certain numbers to algebraic powers. Acta Arith. 32, 63-71. Zbl 357.10018 Bugeaud Y., Gyory K. (1996): Bounds for the solutions of unit equations. Acta Arith. 74, 67-80. Bugeaud Y., Gyory K. (1996): Bounds for the solutions of Thue-Mahler equations and norm-form equations. Acta Arith. 75, 273-292. Bundschuh P. (1969): Arithmetische Untersuchungen unendlicher Produkte. Invent. Math. 6, 275-295. Zbl 195.33703 Bundschuh P. (1970): Ein Satz iiber ganze Funktionen und Irrationalit&saussagen. Invent. Math. 9, 175-184. Zbl 188.10801 Bundschuh P. (1971): Ein Approximationsmass fiir transzendente Losungen gewisser transzendenter Gleichungen. J. Reine Angew. Math. 251, 32-53. Zbl 222.10037 Bundschuh P. (1977/1978): Une nouvelle application de la methode de Gelfond. Semin. Delange-Pisot-Poitou, Theor. Nombres 42, l-8. Zbl 399.10032 Bundschuh P., Hock A. (1969): Bestimmung aller imaginar-quadratischen Zahlkijrper der Klassenzahl Eins mit Hilfe eines Satzes von Baker. Math. Z. 111, 191-204. Zbl 176.33202 of certain Bundschuh P., Shiokawa I. (1984): A measure for the linear independence numbers. Result. Math. 7, 130-144. Zbl 552.10021 Bundschuh P., Waldschmidt M. (1989): Irrationality results for theta functions by Gelfond-Schneider’s method. Acta Arith. 53, 289-307. Zbl 642.10035 Cassels J.W.S. (1950): The rational solutions of the Diophantine equation y2 = x3 - D. Acta Math. 82, 243-273. Zbl 037.02701 Cassels J.W.S. (1957): An Introduction to Diophantine Approximation. Cambridge Univ. Press, Cambridge New York. Zbl 077.04801 Cassels J.W.S. (1960): On a class of exponential equations. Ark. Mat. 4, 231-233. Zbl 102.03507 Catalan E. (1844): Note extraite d’une lettre adressee B l’editeur. J. Reine Angew. Math. 27, 192 Caveny D. (1994): Quantitative transcendence results for numbers associated with Liouville numbers. Proc. Amer. Math. Sot. 120, 349-357 Chebyshev P.L. (1962): Oeuvres. Chelsea, New York (Russ. Orig.: Selected Works, Nauka Moscow 1955. Zbl 064.00108) Cherepnev M.V. (1990): On the algebraic independence of the values of hypergeometric E-functions. Usp. Mat. Nauk 45, No. 6, 155-156. English transl.: Russ. Math. Surv. 45, No. 6, 145-146. Zbl 737.11018 Cherepnev M.V. (1991): On algebraic independence of values of hypergeometric Efunctions satisfying linear non-homogeneous differential equations. Vestn. Mosk. Univ., Ser. I, 46, No. 1, 27-33. English transl.: Most. Univ. Math. Bull. 46, No. 1, 17-21. Zbl 741.11032

References

315

Chirskii V.G. (1973a): Arithmetic properties of values of analytic functions connected by algebraic equations over fields of rational functions. Mat. Zametki 14, 83-94. English transl.: Math. Notes 14, 603-609. Zbl 279.10025 Chirskii V.G. (1973b): Arithmetic properties of the values of elliptic integrals. Vestn. Mosk. Univ., Ser. I, 28, No. 5, 57-64. English transl.: Most. Univ. Math. Bull. 28, No. 5, 48-54. Zbl 281.33001 Chirskii V.G. (1978a): Arithmetic properties of the values of analytic functions with algebraic coefficients in their Taylor series. Vestn. Mosk. Univ., Ser. I, 33, No. 2, 41-47. English transl.: Most. Univ. Math. Bull. 33, No. 2, 32-37. Zbl 386.10019 Chirskii V.G. (1978b): Arithmetic properties of the values of analytic functions with algebraic irrational coefficients of their Taylor’s series. Vestn. Mosk. Univ., Ser. I, 33, No. 3, 29-33. English transl.: Most. Univ. Math. Bull. 33, No. 3, 26-30. Zbl 386.10020 Chirskii V.G. (1978c): Arithmetic properties of the values of hypergeometric functions with irrational parameters. Vestn. Mosk. Univ., Ser. I, 33, No. 5, 3-8. English transl.: Most. Univ. Math. Bull. 33, No. 5, 1-5. Zbl 397.10024 Chirskii V.G. (1979): On the arithmetic properties of the values of elliptic integrals. Vestn. Mosk. Univ., Ser. I, 34, No. 1, 3-11. English transl.: Most. Univ. Math. Bull. 34, No. 1, l-10. Zbl 404.10016 Chirskii V.G. (1990a): On algebraic relations in local fields. Vestn. Mosk. Univ., Ser. I, 45, No. 3, 92-95. English transl.: Most. Univ. Math. Bull. 45, No. 3, 56-58. Zbl 717.11028 Chirskii V.G. (1990b): Global relations. Mat. Zametki 48, 123-127. English transl.: Math. Notes 48, 795-798. Zbl 764.11031 Chirskii V.G. (1992): On algebraic relations in non-archimedean fields. Funkts. Anal. Prilozh. 26, No. 2, 41-50. English transl.: Funct. Anal. Appl. 26, 108-115. Zbl 797.11062 Chirskii V.G., Nurmagomedov M.S. (1973a): On the arithmetical properties of the values of some functions. Vestn. Mosk. Univ., Ser. I, 28, No. 1, 19-25. English transl.: Most. Univ. Math. Bull. 28, No. 1, 16-21. Zbl 254.10032 Chirskii V.G., Nurmagomedov M.S. (1973b): On the arithmetical properties of the values of hypergeometric functions. Vestn. Mosk. Univ., Ser. I, 28, No. 2, 38-45. English transl.: Most. Univ. Math. Bull. 28, No. 1, 69-75. Zbl 265.10020 Chow W.L., van der Waerden B.L. (1937): Zur algebraischen Geometrie. IX. Math. Ann. 113, 692-704. Zbl 016.04004 Chowla S., Cowles J., Cowles M. (1980): Congruence properties of Apery numbers. J. Number Theory 12, 188-190. Zbl 428.10008 Chudakov N.G. (1969): On an upper bound for the discriminant of the tenth imaginary quadratic field of class number 1. Studies in Number Theory. Saratov University, Saratov, 75-77. Zbl 287.12005 Chudakov N.G., Linnik Yu.V. (1950): On a class of completely multiplicative functions. Dokl. Akad. Nauk SSSR 74, 193-196. Zbl 039.03701 Chudnovsky G.V. (1974a): Algebraic independence of some values of the exponential function. Mat. Zametki 15, 661-672. English transl.: Math. Notes 15, 391-398. Zbl 295.10027 Chudnovsky G.V. (1974b): A mutual transcendence measure for some classes’of numbers. Dokl. Akad. Nauk SSSR 218, 771-774. English transl.: Sov. Math., Dokl. 15, 1424-1428. Zbl 305.10029 Chudnovsky G.V. (1974c): Some analytic methods in transcendental number theory. Mathematical Institute Acad. Sciences USSR, Kiev, Preprint No. 8 Chudnovsky G.V. (1974d): Analytic methods in Diophantine approximations. Mathematical Institute Acad. Sciences USSR, Kiev, Preprint No. 9

316

References

Chudnovsky G.V. (1976a): Algebraic independence of constants connected with the exponential and elliptic functions. Dokl. Akad. Nauk Ukr. SSR, No. 8, 697-700. Zbl 337.10024 Chudnovsky G.V. (1976b): The Gelfond-Baker method in problems of Diophantine approximation. Colloq. Math. Sot. Janos Bolyai 13, 19-30. Zbl 337.10023 Chudnovsky G.V. (1977): Towards Schanuel’s conjecture I. Algebraic curves near one point. The general theory of chromatic sequences; and II. Fields of finite transcendence type and chromatic sequences. Stud. Sci. Math. Hung. 12, 125-144 and 145-157. Zbl 442.10023/442.10024 Chudnovsky G.V. (1979a): Independance algebrique des valeurs d’une fonction elliptique en des points algebriques. Formulation des resultats. C.R. Acad. Sci., Paris, Ser. A 288, 439-440. Zbl 408.10023 Chudnovsky G.V. (1979b): Approximations rationnelles des logarithmes de nombres rationnels. CR. Acad. Sci., Paris, Ser. A 288, 607-609. Zbl 408.10019 Chudnovsky G.V. (1979c): Formules d’Hermite pour les approximants de Pade de logarithmes et de fonctions binomes, et mesures d’irrationahte. C.R. Acad. Sci., Paris, Ser. A 288, 965-967. Zbl 418.41011 Chudnovsky G.V. (1979d): Un systeme explicite d’approximants de Pade pour les fonctions hypergeometriques g&CralisCes, avec applications a l’arithmetique. C.R. Acad. Sci., Paris, Ser. A 288, 1001-1004. Zbl 418.41010 Chudnovsky G.V. (1979e): PadC approximations to the generalized hypergeometric functions. J. Math. Pures Appl., IX. Ser. 58, 445-476. Zbl 434.10023 Chudnovsky G.V. (1979f): Transcendental numbers. Lect. Notes Math. 751, 45-69. Zbl 418.10031 Chudnovsky G.V. (1979g): A new method for the investigation of arithmetical properties of analytic functions. Ann. Math., II. Ser. 109, 353-376. Zbl 408.10021 Chudnovsky G.V. (1980a): Algebraic independence of the values of elliptic functions at algebraic points. Elliptic analogue of the Lindemann-Weierstrass theorem. Invent. Math. 61, 267-290. Zbl 434.10024 Chudnovsky G.V. (1980b): Rational and Pade approximations to solutions of linear differential equations and the monodromy theory. Lect. Notes Phys. 126, 136-169. Zbl 457.34003 Chudnovsky G.V. (198Oc): Approximations de PadC explicites pour les solutions des equations differentielles lineaires fuchsiennes. C.R. Acad. Sci., Paris, Ser. A 290, 135-137. Zbl 424.41014 Chudnovsky G.V. (1980d): Independance algebrique dans la methode de GelfondSchneider. C.R. Acad. Sci., Paris, Ser. A 291, 365-368. Zbl 456.10016 Chudnovsky G.V. (1980e): Sur la mesure de la transcendance de nombres dependant d’dquations differentielles de type fuchsien. C.R. Acad. Sci., Paris, Ser. A 291, 485-487. Zbl 452.10035 Chudnovsky G.V. (1980f): Pade approximations and the Riemann monodromy problem. In: Bifurcation Phenomena in Mathematical Physics and Related Topics (Proc. NATO Advanced Study Inst., Cargese, 1979). Reidel, Dordrecht Boston, and Kluwer, Boston, 449-510. Zbl 437.41012 Chudnovsky G.V. (1980g): The inverse scattering problem and applications to arithmetic, approximations theory and transcendental numbers. Lect. Notes Phys. 120, 150-198 Chudnovsky G.V. (1981a): Algebraic grounds for the proof of algebraic independence. How to obtain measure of algebraic independence using elementary methods. 1. Elementary algebra. Commun. Pure Appl. Math. 34, l-28. Zbl 446.10025 Chudnovsky G.V. (1981b): Singular points on complex hypersurfaces and multidimensional Schwarz lemma. SCmin. Delange-Pisot-Poitou, Theor. Nombres 1979/1980 12. Birkhauser, Base1 Boston, 29-69. Zbl 455.32004

References

317

Chudnovsky G.V. (1982a): Measures of irrationality, transcendence and algebraic independence. Lond. Math. Sot. Lect. Note Ser. 56, 11-82. Zbl 489.10027 Chudnovsky G.V. (1982b): Hermite-Pade approximations to exponential functions and elementary estimates of the measure of irrationality of A. Lect. Notes Math. 925, 299-322. Zbl 518.41014 Chudnovsky G.V. (1982c): Criteria of algebraic independence of several numbers. Lect. Notes Math. 9.25, 323-368. Zbl 526.10030 Chudnovsky G.V. (1982d): Recurrences defining rational approximations to irrational numbers. Proc. Japan Acad., Ser. A 58, 129-133. Zbl 503.10022 Chudnovsky G.V. (1983a): On the method of Thue-Siegel. Ann. Math., II. Ser. 117, 325-382. Zbl 518.10038 Chudnovsky G.V. (1983b): Rational approximations to linear forms of exponentials and binomials. Proc. Natl. Acad. Sci. USA 80, 3138-3141. Zbl 511.10027 Chudnovsky G.V. (1983c): Number theoretic applications of polynomials with rational coefficients defined by extremahty conditions. In: Arithmetic and Geometry, Vol. 1. Prog. Math., Boston, Mass. 35, 61-105. Zbl 547.10029 Chudnovsky G.V. (1984a): On some applications of Diophantine approximations. Proc. Natl. Acad. Sci. USA 81, 1926-1930. Zbl 544.10034 Chudnovsky G.V. (1984b): On applications of Diophantine approximations. Proc. Natl. Acad. Sci. USA 81, 7261-7265. Zbl 566.10029 Chudnovsky G.V. (1984c): Contributions to the Theory of Transcendental Numbers. Am. Math. Sot., Providence. Zbl 594.10024 Chudnovsky G.V. (1985a): Pad& approximations to solutions of linear differential equations and applications to Diophantine analysis. Lect. Notes Math. 1052, 85167. Zbl 536.10029 Chudnovsky G.V. (1985b): Applications of Pad6 approximations to Diophantine inequalities in values of G-functions. Lect. Notes Math. 1135, 9-51. Zbl 561.10016 Chudnovsky G.V. (1985c): Applications of Pade approximations to Grothendieck conjecture on linear differential equations. Lect. Notes Math. 1135, 52-100. Zbl 565.14010 Chudnovsky G.V. (1985d): Pade approximations and Diophantine geometry. Proc. Natl. Acad. Sci. USA 82, 2212-2216. Zbl 577.14034 Chudnovsky G.V., Chudnovsky D.V. (1985): Pad& and rational approximations to systems of functions and their arithmetic applications. Lect. Notes Math. 1052, 37-84. Zbl 536.10028 Chudnovsky G.V., Chudnovsky D.V. (1989): Transcendental methods and thetafunctions. Proc. Symp. Pure Math. 49 Part 2, 167-232. Zbl 679.10026 Cijsouw P.L. (1972): Transcendence Measures. Doctoral Dissertation, Univ. Amsterdam. Zbl 252.10031 Cijsouw P.L. (1974): Transcendence measures of certain numbers whose transcendency was proved by A. Baker. Compos. Math. 28, 179-194. Zbl 284.10014 Cijsouw P.L., Tijdeman R. (1975): A n auxiliary result in the theory of transcendental numbers. Duke Math. J. 42, 239-247. Zbl 335.10035 Coates J. (1966): On the algebraic approximation of functions. Indag. Math. 28, 421-461. Zbl 147.03304 Coates J. (1969): An effective p-adic analogue of a theorem of Thue. I. Acta Arith. 15, 279-305. Zbl 221.10025 Coates J. (1970a): An effective padic analogue of a theorem of Thue. II. Acta Arith. 16, 399-412. Zbl 221.10026 Coates J. (1970b): An effective padic analogue of a theorem of Thue. III. Acta Arith. 16, 425-435. Zbl 221.10027 Coates J. (1971a): The transcendence of linear forms in wi, ~2, ~1, 72, 2&. Am. J. Math. 93, 385-397. Zbl 224.10032

318

References

Coates J. (1971b): Linear forms in the periods of the exponential and elliptic functions. Invent. Math. 12, 290-299. Zbl 217.04001 Coates J. (1973): Linear relations between 27ri and the periods of two elliptic curves. In: Diophantine Approximation and Its Applications. Academic Press, London New York, 77-99. Zbl 264.10022 Coates J., Lang S. (1976): Diophantine approximation on abelian varieties with complex multiplication. Invent. Math. 34, 129-133. Zbl 342.10018 Cohen H. (1978): Demonstration de l’irrationalite de C(3) (d’apres R. Apery). Semin. Theor. Nombres Grenoble, 9 p. Cohen P.B. (1995): Proprietes transcendantes des fonctions automorphes, Sbmin. Theor. Nombres Paris 1993-94 Cohen P.B. (to appear): Humbert surfaces and transcendence properties of automorphic functions, Rocky Mt. J. Math. Cohen P.B., Wolfart J. (1990): Modular embeddings for some non-arithmetic Fuchsian groups. Acta Arith. 56, 93-110. Zbl 717.14014 Cohen P.B., Wolfart J. (1993): Fonctions hypergeometriques en plusieurs variables et espaces de modules de varietes abeliennes. Ann. Sci. EC. Norm. Super., IV. Ser. 26, 665-690 Corvaja P. (1992): Roth’s theorem via an interpolation determinant. C.R. Acad. Sci., Paris, Ser. A 315, 517-521. Zbl 784.11030 Cugiani M. (1959a): Sulla approssimabilita di un numero algebrico media&e numeri algebrici di un corpo assegnato. Boll. Unione Mat. Ital. 14, 151-162. ZblO86.26402 Danilov L.V. (1978): Rational approximations of some functions at rational points. Mat. Zametki 24,449-458. English transl.: Math. Notes 24, 741-746. Zbl401.10041 Davenport H. (1968): A note on Thue’s theorem. Mathematika 15, 76-87. Zbl 169.37703 Davenport H., Roth K.F. (1955): Rational approximations to algebraic numbers. Mathematika 2, 160-167. Zbl 066.29302 David S. (1995): Minorations de formes lineaires de logarithmes elliptiques. Mkm. Sot. Math. Fr., Nouv. Ser. 62 Davis C.S. (1978): Rational approximations to e. J. Aust. Math. Sot. 25, 497-502. Zbl 393.10038 Davis C.S. (1979): A note on rational approximation. Bull. Aust. Math. Sot. 20, 407-410. Zbl 407.10026 Debes P. (1986): G-fonctions et theoreme d’irreductibilitk de Hilbert. Acta Arith. 47, 371-402. Zbl 565.12012 Delaunay S. (1992): Grands degres de transcendance, les hypotheses techniques dam. le case exponentiel. I. CR. Acad. Sci., Paris, Ser. A 315, 1333-1336. Zbl 777.11022 Delaunay S. (1993): Grands degres de transcendance, les hypotheses techniques dans le case exponentiel. II. C.R. Acad. Sci., Paris, Ser. A 316, 7-10. Zbl 777.11023 Delone B.N. (1922): Solution of the indefinite equation z3p+y3 = 1. Izv. Ross. Akad. Nauk 16, 273-280. JFM 48.1349.01 Denis L. (1992): Lemmes de zeros et intersections. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 99.104. Zbl 773.14001 Derevyanko N.I. (1989): Approximation of the values of certain functions. Vestn. Mosk. Univ., Ser. I, 44, No. 1, 45-49. English transl.: Most. Univ. Math. Bull. 44, No. 1, 64-70. Zbl 682.10024 Diaz G. (1987): Grands degres de transcendance pour des familles d’exponentielles. C.R. Acad. Sci., Paris, Ser. A 305, 159-162. Zbl 621.10023 Diaz G. (1989): Grands degres de transcendance pour des familles d’exponentielles. J. Number Theory 31, l-23. Zbl 661.10047

References

319

Diaz G. (1990): Une nouvelle mesure. d’independance algebrique pour (a@, #). Acta Arith. 56, 25-32. Zbl 659.10040 Dickson L.E. (1911): On the negative discriminants for which there is a single class of positive primitive binary quadratic forms. Bull. Am. Math. Sot. 17, 534-537. JFM 42.0239.04 Dickson L.E. (1919): History of the Theory of Numbers, Vol. II. Carnegie Institution, Washington. JFM 47.0100.04 Dirichlet P.G. (1840): Sur la theorie des nombres. CR. Acad. Sci., Paris, Ser. A i0, 285-288 Dirichlet P.G. (1842): Veralgemeinerung eines Satzes aus der Lehre von den Kettenbruchen nebst einigen Anwendungen auf die Theorie der Zahlen. Sitzungsber. Preuss. Akad. Wiss., 93-95 Dubitskas A.K. (1987): Approximation of z/a by rational fractions. Vestn. Mosk. Univ., Ser. I, 42, No. 6, 73-76. English transl.: Most. Univ. Math. Bull. 42, No. 6, 76-79. Zbl 635.10026 Dubitskas A.K. (1990a): On approximating certain algebraic numbers and logarithms of algebraic numbers. Candidate’s Dissertation, Moscow State University Dubitskas A.K. (1990b): On approximation of some logarithms of rational numbers by rational fractions of special type. Vestn. Mosk. Univ., Ser. I, 45, No. 2, 69-71. English transl.: Most. Univ. Math. Bull. 45, No. 2, 45-47. Zbl 707.11048 Dubitskas A.K. (1993): Refined approximation to z/d by rational fractions. Vestn. Mosk. Univ., Ser. I, 48, No. 6, 76-77. English transl.: Most. Univ. Math. Bull. 48, No. 6, 44-45. Zbl 844.11045 Durand A. (1980): Simultaneous Diophantine approximations and Hermite’s method: Bull. Aust. Math. Sot. 21, 463-470. Zbl 421.10020 Dvornicich R. (1978): A criterion for the algebraic dependence of two complex numbers. Boll. Unione Mat. Ital. A15, 678-687. Zbl 393.10039 Dvornicich R., Viola C. (1987): Some remarks on Beukers’ integrals. Colloq. Math. Sot. Janos Bolyai 51, 637-657. Zbl 755.11019 Dyson F.J. (1947): The approximation to algebraic numbers by rationais. Acta Math. 79, 225-240. Zbl 030.02101 Easton D. (1986): Effective irrationality measures for certain algebraic numbers. Math. Comput. 46, 613-622. Zbl 586.10019 Edixhoven B., Evertse J.H., eds. (1993): Diophantine Approximation and Abelian Varieties. Lect. Notes Math. 1566, Springer, Berlin Heidelberg New York Ehrenpreis L. (1985): Transcendental numbers and rational differential equations. Lect. Notes Math. 1135, 112-125. Zbl 572.10028 Endell R. (1984): Zur algebraischen Unabhtigigkeit gewisser Werte der Exponentialfunktion. Lect. Notes Math. 1068, 80-85. Zbl 541.10030 Esnault H., Viehweg E. (1984): Dyson’s lemma for polynomials in several variables. Invent. Math. 78, 445-490. Zbl 545.10021 Euler L. (1988): Introduction to Analysis of the Infinite. Springer, Berlin Heidelberg New York. Zbl 657.01013 Evertse J.H. (1983): Upper bounds for the numbers of solutions of Diophantine equations. Math. Centre Tracts, No. 168, 125 p. Zbl 517.10016 Evertse J.H. (1995): An explicit version of FaItings’ product theorem and an improvement of Roth’s lemma. Acta Arith. 73, 215-248 Faddeev D.K. (1934): Complete solution of a class of third degree indeterminate equations of negative discriminant. Proc. 11th All-Union Math. Congress, Leningrad, 2, 36-41 Faisant A., Philibert G. (1987): Quelques resultats de transcendance lies a l’invariant modulaire j, J. Number Theory 25, 184-200 Faltings G. (1991): Diophantine approximations on abelian varieties. Ann. Math., II. Ser. 133, 549-576. Zbl 734.14007

320

References

Faltings G., Wiistholz G. (1994): Diophantine approximations on projective spaces. Invent. Math. 116, 109-138 Fel’dman N.I. (1967a): Lower bound for certain linear forms. Vestn. Mosk. Univ., Ser. Mat., No. 2, 63-72. Zbl 145.05007 Fel’dman N.I. (1967b): Estimation of the absolute value of a linear form from logarithms of certain algebraic numbers. Mat. Zametki 2, 245-256. English transl.: Math. Notes 2, 634-640. Zbl 159.06903 Fel’dman N.I. (1972): On a linear form. Acta Arith. 21, 347-355. Zbl 266.10026 Fel’dman N.I. (1981): Approximation of Algebraic Numbers. Izd. Mosk. Univ., Moscow Fel’dman N.I. (1982): Hilbert’s Seventh Problem. Izd. Mosk. Univ., Moscow. Zbl 515.10031 Fel’dman N.I., Shidlovskij (1967): The development and present state of the theory of transcendental numbers. Usp. Mat. Nauk 22, No. 3, 3-81. English transl.: Russ. Math. Surv. 22, No. 3, l-79. Zbl 178.04801 Flach M. (1989): Periods and special values of the hypergeometric series. Proc. Cambridge Philos. Sot. 106, 389-401 Flicker Y.Z. (1979): Algebraic independence by a method of Mahler. J. Aust. Math. Sot. 27, 173-188. Zbl 399.10035 Franklin P. (1937): A new class of transcendental numbers. Trans. Am. Math. Sot. 42, 155-182. Zbl 017.15205 Galochkin A.I. (1968): Estimate of the conjugate transcendence measure for the values of E-functions. Mat. Zametki 3, 377-386. English transl.: Math. Notes 3, 240-246. Zbl 164.35402 Galochkin A.I. (1970a): A lower bound for linear forms in values of certain hypergeometric functions. Mat. Zametki 8, 19-28. English transl.: Math. Notes 8, 478-484. Zbl 219.10035 Galochkin AI. (1970b): The algebraic independence of values of E-functions at certain transcendental points. Vestn. Mosk. Univ., Ser. I, 25, No. 5, 58-63. English transl.: Most. Univ. Math. Bull. 25, No. 5, 41-45. Zbl 227.10023 Galochkin A.I. (1974): Estimates from below of polynomials in the values of analytic functions of a certain class. Mat. Sb. 95 (137), 396-417. English transl.: Math. USSR, Sb. 24, 385-407. Zbl 311.10035 Galochkin A.I. (1975): Lower bounds of linear forms of values of G functions. Mat. Zametki l&541-552. English transl.: Math. Notes 18,911-917. Zbl319.10039 Galochkin A.I. (1976a): The sharpening of the bounds for certain linear forms. Mat. Zametki 20, 35-45. English transl.: Math. Notes 2U, 575-581. Zbl 336.10028 Galochkin A.I. (1976b): Arithmetic properties of the values of certain entire hypergeometric functions. Sib. Mat. Zh. 17, 1220-1235. English transl.: Sib. Math. J. 17, 894-906. Zbl 348.10025 Galochkin A.I. (1978-79): On Diophantine approximations of the values of some entire functions with algebraic coefficients. Vestn. Mosk. Univ., Ser. I, 33, No. 6, 25-32 (1978) and 34, No. 1, 26-30 (1979). English transl.: Most. Univ. Math. Bull. 33, No. 6, 20-27 and 34, No. 1, 26-31. Zbl 413.10027/413.10028 Galochkin A.I. (1981): Hypergeometric Siegel functions and E-functions. Mat. Zametki 29, 3-14. English transl.: Math. Notes 29, 3-8. Zbl 456.10017 Galochkin AI. (1983): Sur des estimations precises par rapport a la hauteur de certaines formes lineaires. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhiuser, Base1 Boston, 95-98. Zbl 521.10027 Galochkin A.I. (1984): On estimates, unimprovable with respect to height, of some linear forms. Mat. Sb. 124 (166), 416-430. English transl.: Math. USSR, Sb. 52, 407-419. Zbl 559.10028

References

321

Galochkin A.I. (1988): On effective bounds for certain linear forms. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 207-214. Zbl 666.10022 Galochkin AI. (1992): On the solutions of certain equations containing E-functions. Vestn. Mosk. Univ., Ser. I, 47, No. 3, 22-27. English transl.: Most. Univ. Math. Bull. 47, No. 3, 22-26. Zbl 767.11031 Gauss C.F. (1863): Carl Friedrich Gauss Werke, Vol. 3. GSttingen Gel’fond A.O. (1929a): Sur les propi&& arithmetiques des fonctions entieres. Tohoku Math. J., II. Ser. 30, 280-285. JFM 55.0116.01 Gel’fond A.O. (1929b): Sur les nombres transcendantes. CR. Acad. Sci., Paris, Ser. A 189, 1224-1228. JFM 55.0116.02 Gel’fond A.O. (1930): Essay on the history and current state of transcendental number theory. Estestvoznanie i Marksizm 1 (5), 33-55 Gel’fond A.O. (1933): On functions that take integer values at points of a geometric progression. Mat. Sb. 40, 42-47. Zbl 007.12102 Gel’fond A.O. (1934a): On Hilbert’s seventh problem. Dokl. Akad. Nauk SSSR 2, 1-6. Zbl 009.05302 Gel’fond A.O. (1934b): Sur le septieme probleme de Hilbert. Izv. Akad. Nauk SSSR, Ser. Mat. 7, 623-630. Zbl 010.39302 Gel’fond A.O. (1935): On approximating transcendental numbers by algebraic numbers. Dokl. Akad. Nauk SSSR 2, 177-182. Zbl 011.33901 Gel’fond A.O. (1939): On approximating by algebraic numbers the ratio of logarithms of two algebraic numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 3, 509-518. Zbl 024.25103 Gel’fond A.O. (1940): Sur la divisibilite de la difference des puissances de deux nombres entieres par une puissance d’un ideal premier. Mat. Sb. 7 (49), 7-26. Zbl 023.10405 Gel’fond A.O. (1948): ,Approximation of algebraic irrationalities and their logarithms. Vestn. Mosk. Univ., No. 9, 3-25 Gel’fond A.O. (1949a): New investigations on transcendental numbers. Vestn. Akad. Nauk SSSR, No. 11, 98-99 Gel’fond A.O. (194913): On algebraic independence of algebraic powers of algebraic numbers. Dokl. Akad. Nauk SSSR 64, 277-280. Zbl 039.04303 Gel’fond A.O. (1949c): Approximation of algebraic numbers by algebraic numbers, and transcendental number theory. Usp. Mat. Nauk 4, 19-49. Zbl 038.03001 Gel’fond A.O. (1949d): On algebraic independence of certain classes of transcendental numbers. Usp. Mat. Nauk 4, 14-48. Zbl 039.04401 Gel’fond A.O. (1956): Transcendental Numbers. Proceedings of the Third All-Union Math. Congress, Moscow: Akad. Nauk SSSR, 5-6 Gel’fond A.O. (1960): Transcendental and Algebraic Numbers, Dover, New York. Zbl 090.26103 Gel’fond A.O. (1966): Transcendence of certain classes of numbers, in: Gel’fond A.O., Linnik Yu. V.: Elementary Methods in the Analytic Theory of Numbers. MIT Press, Cambridge, Mass. Zbl 142.01403 Gel’fond A.O. (1969): On Hilbert’s seventh problem, in: Hilbert’s Problems. Izd. Nauka, Moscow, 121-127. Zbl 213.28801 Gel’fond A.O. (1973): Selected Works. Izd. Nauka, Moscow. Zbl 275.01022 Gel’fond A.O., Linnik Yu. V. (1948): On Thue’s method and the problem of effectivization in quadratic fields. Dokl. Akad. Nauk SSSR 61, 773-776. Zbl 040.31001 Gel’fond A.O., Fel’dman N.I. (1950): On the relative transcendence measure of certain numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 14, 493-500. Zbl 038.19301 Gessel I. (1982): Some congruences for the Apery numbers. J. Number Theory 14, 362-368. Zbl 482.10003

322

References

Goldfeld D. (1976): The class numbers of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. SC. Norm. Super. Pisa 3, 623-663 Gorelov V.A. (1981): Algebraic independence of the values of some Efunctions. Vestn. Mosk. Univ., Ser. I, 36, No. 1, 47-51. English transl.: Most. Univ. Math. Bull. 36, No. 1, 55-59. Zbl 465.10028 Greminger H. (1930/31): Sur le nombre e. Enseign. Math., II. Ser. 29, 255-259. JFM 57.0242.04 Gross B.H., Zagier D. (1986): Heegner points and derivatives of L-series. Invent. Math. 84, 225-320 Gutnik L.A. (1982): The linear independence over Q of dilogarithms at rational points. Usp. Mat. Nauk 37, No. 5, 179-180. English transl.: Russ. Math. Surv. 37, No. 5, 176-177. Zbl 509.10026 Gutnik L.A. (1983): The irrationality of certain quantities involving C(3). Usp. Mat. Nauk 34 (1979) No. 3, 190 (English transl.: Russ. Math. Surv. 34, No. 3, 200), and Acta Arith. 42 (1983) 255-264. Zbl 414.10030/521.10028 Gyory K. (1980): Resultats effectifs sur la representation des entiers par des formes decomposables. Queen’s Pap. Pure Appl. Math. 56, 142 p. Zbl 455.10011 Gyijry K. (1983): Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains. Acta Math. Acad. Sci. Hung. 42, 45-80 GyBry K, Papp Z. (1983): Norm form equations and explicit lower bounds for linear forms with algebraic coefficients. Stud. Pure Math. Mem. Paul Turrin, Budapest, 245-267. Zbl 518.10020 Gyijry K., Tijdeman R., Voorhove M. (1980): On the equation lk +2” +. . .+z” = yk. Acta Arith. 37, 233-240. Zbl 439.10010 Habsieger L. (1991): Linear recurrent sequences and irrationality measures. J. Number Theory 37, 133-145. Zbl 731.11041 Hasse H. (1939): Simultane Approximation algebraischer Zahlen durch algebraische Zahlen. Monatsh. Math. Phys. 48, 205-225. Zbl 021.29603 Hata M. (1990a): Legendre type polynomials and irrationality measures. J. Reine Angew. Math. 407, 99-125. Zbl 692.10034 Hata M. (1990b): On the linear independence of the values of polylogarithmic functions. J. Math. Pures Appl., IX. Ser. 69, 133-173. Zbl 712.11040 Hata M. (1992): Irrationality measures of the values of hypergeometric functions. Acta Arith. 60, 335-347. Zbl 760.11020 Hata M. (1993a): A lower bound for rational approximations to A. J. Number Theory 43, 51-67. Zbl 776.11034 Hata M. (1993b): Rational approximations to rr and some other numbers. Acta Arith. 63, 335-349. Zbl 776.11033 Heegner K. (1952): Diophantische Analysis und Modulfunktionen. Math. Z. 56, 227253. Zbl 049.16202 Heilbronn H., Linfoot E.H. (1934): On the imaginary quadratic corpora of classnumber one. Q. J. Math., Oxf. II. Ser. 5, 293-301. Zbl 010.33703 Heine E. (1857): Auszug eines Schreibens iiber Kettenbriiche. J. Reine Angew. Math. 53, 284-285 Heine E. (1961): Handbuch der Kugelfunktionen. Physica, Wfirzburg. Zbl 103.29304 Hentzelt K. (1923): Zur Theorie der Polynomideale und Resultanten. Math. Ann. 88, 53-79. JFM 48.0094.03 Hermite Ch. (1873): Sur la fonction exponentielle. C.R. Acad. Sci., Paris, Ser. A 77, 18-24, 74-79, 226-233, and 285-293. JFM 05.0248.01 Hermite Ch. (1917): Oeuvres. Gauthier-Villar, Paris. JFM 46.1417.04 Hilbert D. (1893): Uber die Transcendenz der Zahlen e und z. Math. Ann. 43, 216-219. JFM 25.0734.01

References

323

Hirata-Kohno N. (1986): A quantitative version of the Schneider-Lang theorem. Kokyuroku, Kyoto Univ. 599, 24-32 (for entire collection see Zbl 676.10002) Hirata-Kohno N. (1989): Approximation de periodes de fonctions meromorphes. In: Theorie des Nombres, Conf. Intern. Univ. LavaI 1987, Walter de Gruyter, Berlin New York, 386-397. Zbl 683.10026 Hirata-Kohno N. (1990a): Mesures de transcendance pour les quotients de periodes d’integrales elliptiques. Acta Arith. 56, 111-133. Zbl 699.10049 Hirata-Kohno N. (1990b): Formes lineaires d’integrales elliptiques. In: Sdmin. Theor. Nombres, Paris 1988/1989. Birkhluser, Base1 Boston, Prog. Math. 91, 117-140. Zbl 716.11033 Hirata-Kohno N. (1991): Formes lineaires de logarithmes de points algebriques sur les groupes algebriques. Invent. Math. 104, 401-433. Zbl 716.11035 Hirata-Kohno N. (1992): Nouvelles mesures de transcendance lies aux groupes algebriques commutatifs. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 165-172 Hirata-Kohno N. (1993): Approximations simultanees sur les groupes algebriques commutatifs. Compos. Math. 86, 69-96. Zbl 776.11038 Hodge W.V.D., Pedoe D. (1947): Methods of Algebraic Geometry. Cambridge Univ. Press, Cambridge (Reprint: Cambridge 1994). Zbl 796.14001-3 (3 ~01s.) Holzapfel R. (1993): Transcendental ball points of algebraic Picard integrals. Math. Nachr. 161, 7-25 Hurwitz A. (1883): Uber arithmetische Eigenschaften gewisser transcendenter Funktionen. I. Math. Ann. 22, 211-229. JFM 15.0329.01 Hurwitz A. (1888): Uber arithmetische Eigenschaften gewisser transcendenter Funktionen. II. Math. Ann. 32, 583-588. JFM 20.0392.01 Huttner M. (1986): Probleme de Riemann et irrationahte d’un quotient de deux fonctions hypergeometriques de Gauss. CR. Acad. Sci., Paris, Ser. A 302, 603606. Zbl 607.10026 Huttner M. (1987): Irrationahte de certaines integrales hypergeometriques. J. Number Theory 26, 166-178. 617.10026 Huttner M. (1992): Probleme de Riemann efectif et approximants de Pad&Hermite. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 173-193. Zbl 768.11019 Icen O.S. (1959): Uber die Funktionswerte der padischen Reine CF=“=, #(i),“. J. Reine Angew. Math. 202, 100-106. Zbl 087.04402 Inkeri K. (1964): On Catalan’s problem. Acta Arith. 9, 285-290. Zbl 127.27102 Ivankov P.L. (1985): On approximations by algebraic numbers of roots of certain transcendental equations. Diophantine Approximations, Part I, Izd. Mosk. Univ., 36-42. Zbl 636.10028 Ivankov P.L. (1986): Lower bounds for linear forms in the values of hypergeometric functions. Diophantine Approximations, Part II, Izd. Mosk. Univ., 34-41. Zbl 657.10037 Ivankov P.L. (1987): Simultaneous approximations of values of entire functions by numbers in a cubic field. Vestn. Mosk. Univ., Ser. I, 42, No. 3, 53-56. English transl.: Most. Univ. Math. Bull. 42, No. 3, 43-46. Zbl 623.10022 Ivankov P.L. (1991a): Lower bounds for linear forms in the values of Kummer functions with an irrational parameter. Mat. Zametki 49, 55-63. English transl.: Math. Notes 49, 152-157. Zbl 729.11033 Ivankov P.L. (1991b): On arithmetic properties of the values of hypergeometric functions. Mat. Sb. 182, 282-302. English transl.: Math. USSR, Sb. 72, 267-286. Zbl 724.11036

324

References

Ivankov P.L. (1993): On linear independence of the values of entire hypergeometric functions with irrational parameters. Sib. Mat. Zh. 34, 53-62. English transl.: Sib. Math. J. 34, 839-847 Ivankov P.L. (1994): Estimating measures of linear independence of some numbers. Mat. Zametki 55, 59-67. English transl.: Math. Notes 55 Ivankov P.L. (1995): On linear independence of the values of some functions. Fundament. Priklad. Matematika 1, 191-206 Jabbouri E.M. (1986): Mesures d’independance algebriques de valeurs de fonctions elliptiques et abeliennes. C.R. Acad. Sci., Paris, Ser. A 303, 375-378. Zbl598.10044 Jabbouri E.M. (1992): Sur un critkre pour l’independance algebrique de P. Philippon. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 195-202. Zbl 777.11024 Jacobi C.G.J. (1891): Untersuchungen iiber die Differentialgleichung der hypergeometrischen Reine. Gesammelte Werke, Vol. 6. G. Reiner, Berlin, 184-202. JFM 23.0021.01 Jager H. (1964): A multidimensional generalization of the PadC table. Nederl. Akad. Wet., Proc., Ser. AG 7, 193-249. Zbl 127.27801/151.04505 Jones W.B., Thron W.J. (1980): Continued Fractions, Analytic Theory and Applications. Addison-Wesley, Reading, Mass. Zbl 445.30003 Joyce G., Zucker I. (1991): Special values of the hypergeometric series. Proc. Cambridge Philos. Sot. 109, 257-261 Kaufman R.M. (1971): An estimate of linear forms in the logarithms of algebraic numbers in the p-adic metric. Vestn. Mosk. Univ., Ser. I, 26, No. 2, 3-10. English transl.: Most. Univ. Math. Bull. 26, No. 2, 57-63. Zbl 241.10021 Khinchin A.Ya. (1964): Continued Fractions. Chicago Univ. Press, Chicago. Zbl 117.28601 Kholyavka Ya.M. (1987): Some arithmetic properties of constants connected with elliptic functions. Candidate’s Dissertation, Moscow State University Kholyavka Ya.M. (1990a): Approximation of invariants of the elliptic function. Ukr. Mat. Zh. 42, 681-685. English transl.: Ukr. Math. J. 42, 602-606. Zbl 707.11051 Kholyavka Ya.M. (1990b): On the approximation of numbers connected with elliptic functions. Mat. Zametki 47, 110-118. English transl.: Math. Notes 47, 601-607. Zbl 782.11019 Ko Chao (1965): On the Diophantine equation z* = y” + 1, zy # 0. Sci. Sin., Ser. A 14, 457-460. Zbl 163.04004 Kijhler G. (1980): Some more predictable continued fractions. Monatsh. Math. 89, 95-100. Zbl 419.10010 Koksma J.F. (1935): Ein mengentheoretischer Satz iiber die Gleichverteilung modulo Eins. Compos. Math. 2, 250-258. Zbl 012.01401 Koksma J.F. (1936): Diophantische Approximationen. Berlin. Zbl 012.39602 Koksma J.F., Popken J. (1932): Zur Transcendenz von e”. J. Reine Angew. Math. 168, 211-230. Zbl 005.34902 Kolchin E.R. (1968): Algebraic groups and algebraic dependence. Am. J. Math. 90, 1151-1164. Zbl 169.36701 Kollar J. (1988): Sharp effective Nullstellensatz. J. Am. Math. Sot. 1, 963-975. Zbl 682.14001 Korobov A.N. (1981): Estimates for certain linear forms. Vestn. Mosk. Univ., Ser. I, No. 6, 36-40. English transl.: Most. Univ. Math. Bull. 36, No. 6, 45-49. Zbl 477.10028 Korobov A.N. (1990): Continued fractions and Diophantine approximation. Candidate’s Dissertation, Moscow State University Korobov A.N. (1993): Multidimensional continued fractions and estimates of linear forms. Acta Arith. 71, 331-349

References

325

Krull W. (1948): Parameterspeziahsierung in Polynomringen. II. Arch. Math. 1, 129-137. Zbl 034.1680 Kubota K.K. (1977a): On a transcendence problem of K. Mahler. Can. J. Math. 29, 638-647. Zbl 343.10017 Kubota K.K. (1977b): On the algebraic independence of holomorphic solutions of certain functional equations and their values. Math. Ann. 227, 9-50. Zbl359.10030 Kuz’min R.O. (1930): On a new class of transcendental numbers. Izv. Akad. Nauk SSSR 3, 585-597. JFM 56.0898.03 Lagrange J.L. (1867a): Additions au mkmoire sur la resolution des equations numkriques. Oeuvres, Vol. II. Gauthier-Villars, Paris Lagrange J.L. (1867b): Additions aux elements d’algkbre d’Euler. Oeuvres, Vol. VII. Gauthier-Villars, Paris Lambert J.H. (1946): Opera Mathematics. Turici, Zurich. Zbl 060.01206 Lang S. (1962): A transcendence measure for E-functions. Mathematika 9, 157-161. Zbl 108.26803 Lang S. (1965): Report on Diophantine approximations. Bull. Sot. Math. Fr. 93, 177-192. Zbl 135.10802 Lang S. (1966): Introduction to Transcendental Numbers. Zbl 144.04101 AddisonWesley, Reading, Mass. Lang S. (1971): Transcendental numbers and Diophantine approximations. Bull. Am. Math. Sot. 77, 635-677. Zbl 218.10053 Lang S. (1975): Diophantine approximation on abelian varieties with complex multiplication. Adv. Math. 17, 281-336. Zbl 306.14019 Lang S. (1978): Elliptic Curves: Diophantine Analysis. Springer, Berlin Heidelberg New York. Zbl 388.10001 Lang S. (1984): Algebra. Addison-Wesley, Menlo Park, Cahf. Zbl 712.00001 Langevin M. (1975/1976): Quelques applications de nouveaux resultats de van der Poorten. Sdmin. Delange-Pisot-Poitou, Thkor. Nombres 17, Exp. G12, 11 p. Zbl 354.10008 Langevin M. (1992): Partie sans facteur csrre d’un produit d’entiers voisins. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 203-214. Zbl 776.11058 Lardon R. (1982/1983): Sur les logarithmes de nombres rationnels. Semin. Arith. St.Etienne, VI Laurent M. (1978): Independance algebrique de nombres de Liouville 61eves a des puissances algebriques. C.R. Acad. Sci., Paris, Ser. A 286, 131-133. Zbl 376.10028 Laurent M. (1980a): Transcendance de periodes d’integrales elliptiques. J. Reine Angew. Math. 316, 122-139. Zbl419.10034 Laurent M. (1980b): Independance lindaire de valeurs de fonctions doublement quasiperiodiques. C.R. Acad. Sci., Paris, Ser. A 290, 397-399. Zbl 419.10035 Laurent M. (198Oc): Approximation diophantienne de valeurs de la fonction Beta aux points rationnels. Ann. Fat. Sci. Toulouse, V. Ser. 2, 53-65. Zbl 439.10022 Laurent M. (1983): Sur la transcendance du rapport de deux intkgrales euleriennes. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhauser, Base1 Boston, 133-141. Zbl 514.10028 Laurent M. (1989): Equations exponentielles-polynomes et suites recurrentes lindaires. II. J. Number Theory 31, 24-53. Zbl 661.10027 Laurent M. (1991): Sur quelques resultats ¢s de transcendance. Asterisque 198200, 209-230. Zbl 762.11027 Laurent M. (1992): Hauteur de matrices d’interpolation. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 215-238. Zbl 773.11047 Laurent M. (1994): Linear forms in two logarithms and interpolation determinants. Acta Arith. 66, 181-199. Zbl 809.11038

326

References

Laurent M., Mignotte M., Nesterenko Yu. (1995): Formes lineaires en deux logarithmes et dCterminants d’interpolation. J. Number Theory 55, 285-321. Zbl 843.11036 Lebesgue V.A. (1850): Sur l’impossibilite en nombres entiers de l’equation zm = y2 + 1. Nouv. Ann. Math. 9, 178-181 Legendre A.M. (1860): Elements de geometric, note 4. F. Didot, Paris Lehmer D.H. (1933): On imaginary quadratic fields whose class number is unity. Bull. Am. Math. Sot. 39, 360 LeVeque W.J. (1961): Topics in Number Theory, Vol. 2. Addison-Wesley, Reading, Mass. Zbl 070.03804 Lindemann F. (1882): Uber die Zahl 7~. Math. Ann. 20, 213-225. JFM 14.0369.02 Liouville J. (1840a): Sur l’irrationahte du nombre e. J. Math. Pures Appl. 5, 192 Liouville J. (1840b): Addition a la note sur l’irrationaIit6 du nombre e. J. Math. Pures Appl. 5, 193 Liouville J. (1844a): Sur des classes tres &endues de quantites dont la valeur n’est ni algebrique, ni mtme reductible a des irrationnelles algebriques. C.R. Acad. Sci., Paris, Ser. A 18, 883-885 Liouville J. (1844b): Nouvelle demonstration d’un theoreme sur les irrationnelles algebriques ins&e dans le Compte Rendu de la derniere s&mce. C.R. Acad. Sci., Paris, Ser. A 18, 910-911 Liouville J. (1851): Sur des classes t&s &endues de quantites dont la valeur n’est ni algebrique, ni m&me reductible a des irrationnelles algebriques. J. Math. Pures Appl. 16, 133-142 Lomov IS. (1993): Small denominators in the analytic theory of degenerate differ: ential equations. Differ. Uravn. 29, 2079-2089. English transl.: Differ. Equations 29, 1811-1820. Zbl 824.34011 Lototskii A.V. (1943): Sur l’irrationahte d’un produit infini. Mat. Sb. 12 (54), 262272. Zbl 063.03644 Loxton J.H. (1988): Automata and transcendence. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 215-228. Zbl 656.10032 Loxton J.H., van der Poorten A.J. (1977a): Arithmetic properties of certain functions in several variables. Part I, J. Number Theory 9, 87-106, Zbl 339.10026; Part II, J. Aust. Math. Sot. 24, 393-408, Zbl 371.10027; Part III, Bull. Aust. Math. Sot. 16, 15-47, Zbl 339.10028 Loxton J.H., van der Poorten A.J. (1977b): Tr anscendence and algebraic independence by a method of Mahler. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, 211-226. Zbl 378.10020 Loxton J.H., van der Poorten A.J. (1978): Algebraic independence properties of the Fredholm series. J. Aust. Math. Sot. 26, 31-45. Zbl 392.10034 Loxton J.H., van der Poorten A.J. (1982): Arithmetic properties of the solutions of a class of functional equations. J. Reine Angew. Math. 330, 159-172. Zbl 468.10019 Luke Y.L. (1969): The Special Functions and Their Approximations. Academic Press, London New York. Zbl 193.01701 Lyapunov A.M. (1959): Collected Works, Vol. 4. Izd. Akad. Nauk SSSR, Moscow. Zbl 192.31204 Mahler K. (1929): Arithmetische Eigenschaften der Losungen einer Klasse von Funktionalgleichungen. Math. Ann. 101, 342-366. JFM 55.0115.01 Mahler K. (1930a): Uber das Verschwinden von Potenzreihen mehrerer Veranderlichen in speziahen Punktfolgen. Math. Ann. 103, 573-587. JFM 56.0185.03 Mahler K. (1930b): Arithmetische Eigenschaften einer Klasse transzendental - transcendenter Funktionen. Math. Z. 32, 545-585. JFM 56.0186.01

References

327

Mahler K. (1931): Ein Beweis des Thue-Siegelschen Satzes iiber die Approximation aigebraischer Zahlen fiir binomische Gleichungen. Math. Ann. 105, 267-276. Zbl 002.18401 Mahler K. (1932a): Zur Approximation der Exponentialfunktion und des Logarithmus. J. Reine Angew. Math. 166, 118-136 and 137-150. Zb1003.15101/003.38805 Mahler K. (1932b): Ein Beweis der Transzendenz der p-adischen Exponentialfunktion. J. Reine Angew. Math. 169, 61-66. Zbl 006.01101 Mahler K. (1933): Zur Approximation algebraischen Zahlen. Math. Ann. 107, 691-730 and 108, 37-55; and Acta Math. 62, 91-166. Zbl 006.10502/006.15604/ 008.19801 Mahler K. (1935): Uber transzendente padische Zahlen. Compos. Math. 2, 259-275. Zbl 012.05302 Mahler K. (1936): Ein Analogon zu einem Schneiderschen Satz. Proc. Kon. Akad. Wetensch. A. 39, 633-640 and 729-737. Zbl 014.20502/014.20503 Mahler K. (1937): Arithmetische Eigenschaften einer Klasse von Dezimalbruchen. Proc. Kon. Akad. Wetensch. A. 40, 421-428. Zbl 017.05602 Mahler K. (1953a): On the approximation of ‘IT. Proc. Kon. Akad. Wetensch. A. 56, 30-42. Zbl 053.36105 Mahler K. (1953b): On the approximation of logarithms of algebraic numbers. Phi10s. Trans. R. Sot. Lond., Ser. A 245, 371-398. Zbl 052.04404 Mahler K. (1957): On the fractional parts of powers of a rational number. II. Mathematika 4, 122-124. Zbl 208.31002 Mahler K. (1961): Lectures on Diophantine Approximations, Part I. Cushing-Malloy, Inc., Ann Arbor, Mich. Zbl 158.29903 Mahler K. (1967): Applications of some formulae by Hermite to the approximation of exponentials and logarithms. Math. Ann. 168, 200-227. Zbl 144.29201 Mahler K. (1968a): Perfect systems. Compos. Math. 19, 95-166. Zbl 168.31303 Mahler K. (1968b): Applications of the theorem by A.B. Shidlovskij. Proc. R. Sot. Lond., Ser. A 305, 149-173. Zbl 164.05702 Mahler K. (1976): Lectures on Transcendental Numbers. Lect. Notes Math. 546, Springer, Berlin Heidelberg New York. Zbl 332.10019 Maier W. (1927): Potenzreihen irrationalen Grenzwertes. J. Reine Angew. Math. 156, 93-148. JFM 53.0340.02 Maillet E. (1906): Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions. Gauthier-Villar, Paris. JFM 37.0237.02 Maillet E. (1916): Sur un theoreme de M. Axe1 Thue. Nouv. Ann. Math., Ser. 4 16, 338-345. JFM 46.0285.02 Makarov Yu.N. (1978): Estimates of the measure of linear independence of the values of E-functions. Vestn. Mosk. Univ., Ser. I, 33,No. 2, 3-12. English transl.: Most. Univ. Math. Bull. 33, No. 2, l-8. Zbl 386.10017 Makarov Yu.N. (1984): On estimates of linear forms of the values of E-functions. Vestn. Mosk. Univ., Ser. I, 39, No. 2, 37-42. English transl.: Most. Univ. Math. Bull. 39, No. 2, 51-56. Zbl 537.10022 Markov A.A. (1948): Selected Works on Continued Fractions and the Theory of Functions of Least Deviation from Zero. Gos. Izd. Tekhniko-Teoret. Lit., Moscow. Zbl 034.03703 Masser D. (1975): Elliptic Functions and Transcendence. Lect. Notes Math. 437, Springer, Berlin Heidelberg New York. Zbl 312.10023 Masser D. (1976): Linear forms in algebraic points of abelian functions. III. Proc. Lond. Math. Sot. 33, 549-564. Zbl 334.14019 Masser D. (1977a): The transcendence of certain quasi-periods associated with abelian functions in two variables. Compos. Math. 35, 239-258. Zbl 371.10026

328 Masser

References D. (1977b):

Some vector

spaces associated

with

two elliptic

functions.

In:

lkanscendence Theory:AdvancesandApplications.AcademicPresqLondonNew

York, 101-119. Zbl 362.10029 Masser D. (1979): Some recent results in transcendence theory. Asterisque 61, 145154. Zbl 402.10036 Masser D. (1981): On polynomials and exponential polynomials in several complex variables. Invent. Math. 63, 81-95. Zbl 454.10019 Masser D. (1984): Zero estimates on group varieties. Proc. Int. Congr. Math., Warsaw, Vol. 1, 493-502. Zbl 564.10040 Masser D., Wiistholz. G. (1981): Zero estimates on group varieties. I. Invent. Math. 64, 489-516. Zbl 467.10025 Masser D., Wiistholz G. (1982): Algebraic independence properties of values of elliptic functions. Lond. Math. Sot. Lect. Note Ser. 56, 360-363. Zbl 491.10025 Masser D., Wiistholz G. (1983): Fields of large transcendence degree generated by values of elliptic functions. Invent. Math. 72, 407-463. Zbl 516.10027 Masser D., Wiistholz G. (1985): Zero estimates on group varieties. II. Invent. Math. 80, 233-267. Zbl 564.10041 Masser D., Wiistholz G. (1986): Algebraic independence of values of elliptic functions. Math. Ann. 276, 1-17. Zbl 597.10034 Masser D., Wiistholz G. (1990): Estimating isogenies on elliptic curves. Invent. Math. 100, l-24 Masser D., Wiistholz G. (1993a): Periods and minimal abelian subvarieties. Ann. Math., II. Ser. 137, 407-458 ,Masser D., Wiistholz G. (1993b): Isogeny estimates for abelian varieties, and finiteness theorems. Ann. Math., II. Ser. 137, 459-472 Masser D., Wiistholz G. (1993c): Galois properties of division fields of elliptic curves. Bull. Lond. Math. Sot. 25, 247-254 Masser D., Wiistholz G. (1994): Endomorphism estimates for abelian varieties. Math. Z. 215, 641-653 Masser D., Wiistholz G. (1995a): Factorization estimates for abelian varieties. Publ. Math., Inst. Hautes Stud. Sci., No. 81, l-24 Masser D., Wiistholz G. (199513): Refinements of the Tate conjecture for abelian varieties. In: Barth W., Hulek K., Lange H., eds., Abelian Varieties, Proc. Intern. Conf. Egloffstein, Germany 1993. Walter de Gruyter, Berlin New York, 211223 Matala-Aho T., VLLnanen K. (1981): On the arithmetic properties of certain values of one Gauss hypergeometric function. Acta Univ. Oulu., Ser. A 112, 25 p. Zbl 485.10025 Matveev E.M. (1979): Some Diophantine equations with a large number of variables. Vestn. Mosk. Univ., Ser. I, 34, No. 2, 22-30. English transl.: Most. Univ. Math. Bull. 34, No. 2, 22-31. Zbl 409.10010 Matveev E.M. (1980): Some Diophantine equations with effective bounds for the modulus of continuity. Vestn. Mosk. Univ., Ser. I, 35, No. 6, 23-27. English transl.: Most. Univ. Math. Bull. 35, No. 6, 23-27. Zbl 452.10020 Matveev E.M. (1981): Estimates for linear forms in values of G-functions and their applications to the solution of Diophantine equations. Usp. Mat. Nauk 36, No. 2, 191-192. English transl.: Russ. Math. Surv. 36, No. 2, 183-184. Zbl 461.10032 Matveev E.M. (1982): Linear forms in the values of G-functions and Diophantine equations. Mat. Sb. 117 (159), 379-396. English transl.: Math. USSR, Sb. 45, 379-396. Zbl 492.10028 Matveev E.M. (1992): On fundamental units of certain fields. Mat. Sb. 183, 35-48. English transl.: Russ. Acad. Sci. Sb. Math. 76, 293-304. Zbl 768.11050

References

329

Mignotte M. (1974): Approximations rationnelles de z et quelques autres nombres. Bull. Sot. Math. Fr. 37, 121-132. Zbl 286.10017 Mignotte M. (1975): Independance algebrique de certains nombres de la forme crp et cyp2. SCmin. D&urge-Pisot-Poitou, Theor. Nombres 16, 9/l-9/5. Zbl 324.10032 Mignotte M. (1977): Approximation des Nombres Algebriques. Univ. Paris-Sud, Orsay. Zbl 399.10033 Mignotte M. (1992): Sur l’bquation de Catalan. I. C.R. Acad. Sci., Paris, Ser. A 314, 165-168. Zbl 749.11025 Mignotte M., Waldschmidt M. (1978): Linear forms in two logarithms and Schneider’s method. I. Math. Ann. 231, 241-267. Zbl 361.10027 and SchneiMignotte M., Waldschmidt M. (1989a): L’mear forms in two logarithms der’s method. II. Acta Arith. 53, 251-287. Zbl 642.10034 Mignotte M., Waldschmidt M. (1989b): Linear forms in two logarithms and Schneider’s method. III. Ann. Fsc. Sci. Univ. Toulouse, V. SBr, Math. 97, 43-75. Zbl 702.11044 Miller W. (1982): Transcendence measures by a method of Mahler. J. Aust. Math. Sot. 32, 68-78. Zbl 482.10036 Mimura Y. (1983): Congruence properties of Apery numbers. J. Number Theory 16, 138-146. Zbl 504.10007 Minkowski H. (1910): Geometrie der Zahlen. Teubner, Leipzig (Reprint: 1925, JFM 51.0161.01; 1953 Chelsea, Zbl 050.04807) Molchanov SM. (1985): Estimates of the transcendence measure in Mahler’s method. Diophantine Approximation, Part I, Izd. Mosk. Univ., 56-65. Zbl 645.10032 Mordell L.J. (1923): On the integer solutions of the equation ey2 = a~~+bz~+cz+d. Proc. Lond. Math. Sot. II. Ser. 21, 415-419. JFM 49.0097.02 Mordell L.J. (1969): Diophantine Equations. Academic Press, London New York. Zbl 188.34503 Mordukhai-Boltovskoi D.D. (1927): On certain properties of transcendental numbers of the first class. Mat. Sb. 34, 55-99. JFM 53.0166.03 Moreau J.C. (1983a): Demonstrations geometriques de lemmes de zeros. I. SCmin. DClange-Pisot-Poitou 1981/1982, Theor. Nombres. Birkhauser, Base1 Boston, 201205. Zbl 578.10040 Moreau J.C. (1983b): Demonstrations geometriques de lemmes de zeros. II. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhauser, Base1 Boston, 191-197. Zbl 578.10040 Morita Y. (1972): On transcendency of special values of arithmetic automorphic functions. J. Math. Sot. Japan 24, 268-274 Mueller J. (1984): On Thue’s principle and its applications. Lect. Notes Math. 1068, 158-166. Zbl 539.10027 Nagaev N.D. (1977b): On transcendence of a linear form in wj, qj, Pri, logo. Izv. Voronezh. Gos. Pedagog. Inst. 187, 57-71 Nagell T. (1921): Des equations indeterminees z2 + z + 1 = y” et z2 + z + 1 = 3yn. Norsk. Math. Foren. Skr. 1, No. 2, 1-14. JFM 48.0138.01 Nesterenko Yu.V. (1969): On the algebraic independence of the values of E-functions satisfying nonhomogeneous linear differential equations. Mat. Zametki 5, 587-589. English transl.: Math. Notes 5, 352-358. Zbl 212.39401 Nesterenko Yu.V. (1972): Estimates of the orders of zeros of analytic functions of a certain class and their application to the theory of transcendental numbers. Dokl. Akad. Nauk SSSR 205, 292-295. English transl.: Sov. Math., Dokl. 13, 938942. Zbl 265.30006 Nesterenko Yu.V. (1974a): Order function for almost all numbers. Mat. Zametki 15, 405-414. English transl.: Math. Notes 15, 234-240. Zbl 287.10022

330

References

Nesterenko Yu.V. (1974b): On the algebraic dependence of the components of solutions of a system of linear differential equations. Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495-512. English transl.: Math. USSR, Izv. 8, 501-518. Zbl 308.35016 Nesterenko Yu.V. (1977): Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284. English transl.: Math. USSR, Izv. 11, 239-270. Zbl 354.10026 Nestetenko Yu.V. (1979): On the number ‘IT and Lindemann’s theorem. Commentat. Math. Univ. Carol. 20, 335-343. Zbl 425.10040 Nesterenko Yu.V. (1983a): On the algebraic independence of algebraic numbers to algebraic powers. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhauser, Base1 Boston, 199-220. Zbl 513.10035 Nesterenko Yu.V. (1983b): Estimates of orders of zeros of functions of a certain class. Mat. Zametki 33, 195-205. English transl.: Math. Notes 33, 98-103. Zbl 525.10017 Nesterenko Yu.V. (1983c): On a sufficient criterion for algebraic independence of numbers. Vestn. Mosk. Univ., Ser. I, 38, No. 4, 63-68. English transl.: Most. Univ. Math. Bull. 38, No. 4, 70-76. Zbl 524.10027 Nesterenko Yu.V. (1984a): Estimates for the characteristic function of a prime ideal. Mat. Sb. 123 (165), 11-34. English transl.: Math. USSR, Sb. 51, 9-32. Zbl 579.10030 Nesterenko Yu.V. (1984b): On algebraic independence of algebraic powers of algebraic numbers. Mat. Sb. 123 (165), 435-459. English transl.: Math. USSR, Sb. 51, 429-454. Zbl 549.10023 Nesterenko Yu.V. (1985a): On a measure of the algebraic independence of the values of certain functions. Mat. Sb. 128 (170), 545-568. English transl.: Math. USSR, Sb. 56, 545-567. Zbl 603.10033 Nesterenko Yu.V. (1985b): On the linear independence of numbers. Vestn. Mosk. Univ., Ser. I, 40, No. 1, 46-49. English transl.: Most. Univ. Math. Bull. 40, No. 1, 69-74. Zbl 572.10027 Nesterenko Yu.V. (1985c): On estimating the algebraic independence measure of numbers. Diophantine Approximation, Part I, Izd. Mosk. Univ., 65-77. Zbl 649.10024 Nesterenko Yu.V. (1985d): On the measure of algebraic independence of the values of an elliptic function at algebraic points. Usp. Mat. Nauk 40, No. 4, 221-222. English transl.: Russ. Math. Surv. 40, No. 4, 237-238. Zbl 603.10034 Nesterenko Yu.V. (1987): On the number w. Vestn. Mosk. Univ., Ser. I, 42, No. 3, 7-10. English transl.: Most. Univ. Math. Bull. 42, No. 3, 6-10. Zbl 622.10024 Nesterenko Yu.V. (1988a): Estimates for the number of zeros of certain functions. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 263-269. Zbl 656.10034 Nesterenko Yu.V. (1988b): Effective estimates of algebraic independency measure of values of E-functions. Vestn. Mosk. Univ., Ser. I, 43, No. 4, 85-88. English transl.: Most. Univ. Math. Bull. 43, NO. 4, 84-87. Zbl 659.10042 Nesterenko Yu.V. (1989a): Transcendence degree of some fields generated by values of the exponential function. Mat. Zametki 46, 40-49. English transl.: Math. Notes 46, 706-711. Zbl 691.10030 Nesterenko Yu.V. (1989b): Estimates for the numbers of zeros of certain classes of functions. Acta Arith. 53, 29-46. Zbl 717.11026 Nesterenko Yu.V. (1992): On a measure of the algebraic independence of the values of elliptic functions. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 239-248. Zbl 786.11047 Nesterenko Yu.V. (1993): Hermite-Pa& approximations of the generalized hypergeometric functions. In: Semin. Theor. Nombres, Paris 1991/1992. Prog. Math. 116, 191-216. Zbl 824.41017

References

331

Nesterenko Yu.V. (1994): Hermite-Pade approximants of generalized hypergeometric functions. Mat. Sb. 185, 39-72. English transl.: Russ. Acad. Sci. Sb. Math. 83 (1995) 189-219 Nesterenko Yu.V. (1995): On the algebraic independence measure of the values of elliptic functions. Izv. Akad. Nauk SSSR, Ser. Mat. 59, 155-178 Nesterenko Yu.V. (1996a): Modular functions and transcendence problems. C.R. Acad. Sci., Paris, Ser. A 322, 909-914 Nesterenko Yu.V. (1996b): Modular functions and transcendence problems. Mat. Sb. 187, 65-96 Nesterenko Yu.V., Shidlovskij A. (1996): On the linear independence of values of E-functions. Mat. Sb. 187 Nguyen Tien Dai (1983): On estimates for the orders of zeros of polynomials in analytic functions and their application to estimates for the relative transcendence measure of values of E-functions. Mat. Sb. 120 (162), 112-142. English transl.: Math. USSR, Sb. 48, 111-140. Zbl 516.10025 Nikishin E.M. (1979a): On irrationality of the values of the functions F(z,s). Mat. Sb. 109 (151), 410-417. English transl.: Math. USSR, Sb. 37, 381-388. Zbl 414.10032 Nikishin E.M. (1979b): On logarithms of natural numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 43, 1319-1327, and 44, 972. English transl.: Math. USSR, Izv. 15, 523530. Zbl 422.10024 Nikishin E.M. (1979c): A system of Markov functions. Vestn. Mosk. Univ., Ser. I, 34, No. 4, 60-63. English transl.: Most. Univ. Math. Bull. 34, No. 4, 63-66. Zbl 425.30006 Nikishin E.M. (1980): On simultaneous Pad& approximants. Mat. Sb. 113 (155), 499-519. English transl.: Math. USSR, Sb. 41, 409-425. Zbl 456.30009 Nikishin E.M. (1982): Arithmetic properties of the Markov function for the Jacobi weight. Anal. Math. 8, 39-46. Zbl 495.10022 Nikishin E.M., Sorokin V.N. (1991): Rational Approximations and Orthogonality. Am. Math. Sot., Providence, Rhode Island. Zbl 733.41001 Nishioka K. (1986): Algebraic independence of certain power series of algebraic numbers. J. Number Theory 23, 354-364. Zbl 589.10035 Nishioka K. (1990a): On an estimate for the orders of zeros of Mahler type functions. Acta Arith. 56, 249-256. Zbl 719.11041 Nishioka K. (1990b): New approach in Mahler’s method. J. Reine Angew. Math. 407, 202-219. Zbl 694.10035 Nishioka K. (1991): Algebraic independence measures of the values of Mahler functions. J. Reine Angew. Math. &?O, 203-214. Zbl 736.11034 Nishioka K., Topfer T. (1991): Transcendence measures and nonlinear functional equations of Mahler type. Arch. Math. 57, 370-378. Zbl 746.11030 Noether E. (1923): Eliminationstheorie und allgemeine Idealtheorie. Math. Ann. 90, 229-261 JFM 49.0076.04 Nurmagomedov MS. (1971): Arithmetic properties of the values of a class of analytic functions. Mat. Sb. 85 (127), 339-365. English transl.: Math. USSR, Sb. 14, 329357. Zbl 225.10037 Oleinikov V.A. (1962): On transcendence and algebraic independence of the values of certain E-functions. Vestn. Mosk. Univ., Ser. I, No. 6, 34-38. Zbl 118.28302 Oleinikov V.A. (1966): Algebraic independence of values of E-functions satisfying linear nonhomogeneous differential equations of third order. Dokl. Akad. Nauk SSSR 169, 32-34. English transl.: Sov. Math., Dokl. 7, 869-871. Zbl 163.05001 Oleinikov V.A. (1968): On the transcendence and algebraic independence of the values of certain entire functions. Izv. Akad. Nauk SSSR, Ser. Mat. 32, 63-92. English transl.: Math. USSR, Izv. 2, 61-88. Zbl 212.39203

332

References

Oleinikov V.A. (1969): The algebraic independence of the values of Efunctions. Mat. Sb. 78 (120), 301-306. English transl.: Math. USSR, Sb. 7, 293-298. Zbl 213.06103 Oleinikov V.A. (1970): On differential irreducibility of a linear nonhomogeneous equation. Dokl. Akad. Nauk 194, 1017-1020. English transl.: Sov. Math., Dokl. 11, 1337-1341. Zbl 227.10024 Oleinikov V.A. (1986): Conditions of mutual transcendence for hypergeometric series. Vestn. Mosk. Univ., Ser. I, 41, No. 1, 20-25. English transl.: Most. Univ. Math. Bull. 41, No.1, 26-32. Zbl 596.10034 Osgood C. (1966): Some theorems on Diophantine approximation. Trans. Am. Math. Sot. 123, 64-87. Zbl 152.03607 Osgood C. (1970): The simultaneous Diophantine approximation of certain IE-th roots. Proc. Cambridge Philos. Sot. 67, 75-86. Zbl 191.05401 PadC H. (1901): Sur l’expression g&&ale de la fonction rationnelle approchee de (1 + x)~. C.R. Acad. Sci., Paris, Ser. A 132, 754-756. JFM 32.0436.03 Pade H. (1907): Recherches sur la convergence des developpements en fractions continues d’une certaine categoric de fonctions. Ann. Sci. EC. Norm. Super. 24, 341-400. JFM 38.0265.01 Peck L.G. (1961): Simultaneous rational approximations to algebraic numbers. Bull. Am. Math. Sot. 67, 197-201. Zbl 098.26302 Perron 0. (1951): Irrationalzahlen. Chelsea, New York (1960 deGruyter, Zbl 090.03202; Orig. ed. Goschen 1921, JFM 48.0190.05) Perron 0. (1954,1957): Die Lehre von den Kettenbriichen. Vol. 1, Teubner, Stuttgart; Vol. 2, Teubner, Leipzig. Zbl 056.05901/077.06602 Pershikova T.V. (1966): On transcendence of the values of certain E-functions. Vestn. Mosk. Univ., Ser. I, No. 2, 55-61. Zbl 149.29401 Philibert G. (1988): Une mesure d’independance algebrique. Ann. Inst. Fourier 38, 85-103. Zbl 644.10026 Philippon P. (1981): Independance algebrique de valeurs de fonctions exponentielles’ padiques. J. Reine Angew. Math. 329, 42-51. Zbl 459.10024 Philippon P. (1982): Independance algebrique et varietes abeliennes. C.R. Acad. Sci., Paris, Ser. A 29, 257-259. Zbl 498.14009 Philippon P. (1983): VariCtes abeliennes et independance algkbrique. Invent. Math. 70, 289-318 and 72, 389-405. Zbl 503.10024/516.10026 Philippon P. (1984): Criteres pour l’independance algebrique de familles de nombres. Math. Rep. Acad. Sci. Can. 6, 285-290. Zbl 567.10033 Philippon P. (1985): Sur les mesures d’independance algebrique. SCmin. DClangePisot-Poitou, Theor. Nombres 1983/1984. Birkhauser, Base1 Boston, 219-233. Zbl 567.10034 Philippon P. (1986a): Criteres pour l’independance algebrique. Publ. Math., Iust. Hautes Stud. Sci. 64, 5-52. Zbl 615.10044 Philippon P. (1986b): Lemmes de zeros dans les groupes algebriques commutatifs. Bull. Sot. Math. Fr. 114, 355-383. Errata and addenda in 115, 397-398. Zbl 617.14001/634.14001 Philippon P. (1988a): A propos du texte de W.D. Brownawell: Bounds for the degree in the Nullstellensatz. Ann. Math., II. Ser. 127, 367-371. Zbl 641.14001 Philippon P. (1988b): Theo&me des zeros effectif, d’apres J. Kollk. Publ. Math. Univ. Paris-VI 88, No. 6 Philippon P. (1989): Theo&me des zeros effectif et elimination. Semin. Theor. Nombres Bordx., Ser. II 1, 137-155. Zbl 743.11036 Philippon P. (1991): Denominateurs dans le theoreme des zeros de Hilbert. Acta Arith. 58, l-25. Zbl 717.13015

References

333

Philippon P. (1992): Criteres pour l’indkpendance algebrique dans les anneaux diophantiens. C.R. Acad. Sci., Paris, Ser. A 315, 511-515. Zbl 781.13011 Philippon P., Waldschmidt M. (1988a): Lower bounds for linear forms in logarithms. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 280-312. Zbl 659.10037 Philippon P., Waldschmidt M. (1988b): Formes lineaires de logarithmes simultanees sur les groupes algebriques. Ill. J. Math. 32, 281-314 Polya G. (1915): Uber ganzwertige ganze Funktionen. Rend. Circ. Mat. Palermo 40, 1-16. JFM 45.0655.02 Polya G. (1918): Zur arithmetischen Untersuchungen de Polynome. Math. Z. 1, 143148. JFM 46.0240.04 van der Poorten A. (1971): On the arithmetic nature of definite integrals of rational functions. Proc. Am. Math. Sot. 29, 451-456. Zbl 215.35301 van der Poorten A. (1979): A proof that Euler missed - Apery’s proof of the irrationality of C(3). Math. Intell. 1, 195-203. Zbl 409.10028 Popken J. (1929a): Zur Transzendenz von e. Math. Z. 29, 525-541. JFM 55.0117.01 Popken J. (1929b): Zur Transzendenz von ?r. Math. Z. 29, 542-548. JFM 55.0117.02 Popov A.Yu. (1985): Approximation of certain powers of e. Diophantine Approximation, Part I, Izd. Mosk. Univ., 77-85. Zbl 638.10036 Popov A.Yu. (1986): Arithmetic properties of the values of certain infinite products. Diophantine Approximation, Part II, Izd. Mosk. Univ., 63-78. Zbl 648.10022 Popov A.Yu. (1990): Approximation of values of some infinite products. Vestn. Mosk. Univ., Ser. I, 45, No. 6, 3-6. English transl.: Most. Univ. Math. Bull. 45, No. 6, 4-6. Zbl 722.11036 Radoux C. (1980): Quelques proprietes arithmetiques des nombres d’Ap&y. CR. Acad. Sci., Paris, Ser. A 291, 567-569. Zbl 445.10013 Ramachandra K. (1968): Contributions to the theory of transcendental numbers. Acta Arith. 14, 65-72 and 73-88. Zbl 176.33101 Ramachandra K., Shorey T.N., Tijdeman R. (1975): On Grimm’s problem relating to factorization of a block of consecutive integers. I. J. Reine Angew. Math. 273, 109-124. Zbl 297.10031 Ramachandra K., Shorey T.N., Tijdeman R. (1976): On Grimm’s problem relating to factorization of a block of consecutive integers. II. J. Reine Angew. Math. 288, 192-201. Zbl 333.10028 Reyssat E. (1978/1979): Irrational&e de C(3) selon Apery. Semin. DBlange-PisotPoitou, Theor. Nombres, VI. Zbl 423.10020 Reyssat E. (1980a): Approximation de nombres lies a la fonction sigma de Weierstrass. Ann. Fat. Sci. Univ. Toulouse, V. Ser., Math. 2, 79-91. Zbl 439.10021 Reyssat E. (1980b): Fonctions de Weierstrass et independance algebrique. C.R. Acad. Sci., Paris, Ser. A 290, 439-441. Zbl 426.10037 Reyssat E. (198Oc): Approximation algebrique de nombres lies aux fonctions elliptiques et exponentielle. Bull. Sot. Math. Fr. 108, 47-79. Zbl 432.10018 Reyssat E. (1981): Un critere d’independance algebrique. J. Reine Angew. Math. 329, 66-81. Zbl 459.10023 Reyssat E. (1983): Mesures de transcendance pour les logarithmes de nombres rationnels. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhauser, Base1 Boston, 235-245. Zbl 522.10023 Rhin G. (1983): Sur l’approximation diophantienne simultanee de deux logarithmes de nombres rationnels. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. BirkhBuser, Base1 Boston, 247-258. Zbl 518.10039 Rhin G. (1987): Approximants de Pade et mesures effectives d’irrationahte. Semin. Theor. Nombres, Paris 1985/1986. Birkhauser, Base1 Boston, 155-164. Zbl 632.10034

334

References

Rhin G., Toffin P. (1986): Approximants

de Pade simultanes

de logarithmes.

J. Num-

ber The0r.y 24, 284-297. Zbl 596.10033 Rhin

G., Viola

C. (1993):

On the irrationality

measure

of C(2). Ann. Inst. Fourier

43, 85-109. Zbl 776.11036 Ricci G. (1935): Sul settimo problema di Hilbert. Ann. SC. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 341-372. Zbl 012.24801 Riemann B. (1876): Gesammelte mathematische Werke. Teubner, Leipzig. JEM 08.0231.03 van Rossum H. (1955): Systems of orthogonal and quasiorthogonal polynomials connected with the Pad6 table. Indag. Math. 17, 517-525, 526-534, and 675-682. Zbl 067.29301 Roth K.F. (1955): Rational approximations to algebraic numbers. Mathematika 2, 160-167. Zbl 066.29302 Roy D. (1992): Matrices whose coefficients are linear forms in logarithms. J. Number Theory 41, 22-47. Zbl 763.11030 Rukhadze E.A. (1987): Lower estimate for rational approximations of ln2. Vestn. Mosk. Univ., Ser. I, 42, No. 6, 25-29. English transl.: Most. Univ. Math. Bull. 4.2, No. 6, 30-35. Zbl 794.11028 Salikhov V.Kh. (1977,198O): On the differential irreducibility of a class of differential equations. Dokl. Akad. Nauk SSSR 235, 30-33, and Izv. Akad. Nauk SSSR, Ser. Mat. 44, 176-202. English transl.: Sov. Math., Dokl. 18, 891-895, and Math. USSR, Izv. 16, 165-190. Zbl 389.34008/463.10026 Salikhov V.Kh. (1988): Formal solutions of linear differential equations and their application in the theory of transcendental numbers. Tr. Mosk. Mat. O.-va 51; 223-256. English transl.: Trans. Most. Math. Sot. 51, 219-251. Zbl 679.10027 Salikhov V.Kh. (1989): On algebraic independence of the values of hypergeometric E-functions. Dokl. Akad. Nauk SSSR 307, 284-287. English transl.: Sov. Math., Dokl. 40, 71-74. Zbl 712.11043 Salikhov V.Kh. (1990a): A criterion for the algebraic independence of the values of a class of hypergeometric E-functions. Mat. Sb. 181, 189-211. English transl.: Math. USSR, Sb. 69, 203-226. Zbl 712.11044 Salikhov V.Kh. (1990b): Irreducibility of hypergeometric equations and algebraic independence of values of E-functions. Acta Arith. 53, 453-471. Zbl 831.11037 Schinzel A. (1962): The intrinsic divisors of Lehmer numbers in the case of negative discriminant. Ark. Mat. 4, 413-416. Zbl 106.03105 Schinzel A. (1967): On two theorems of Gelfond and some of their applications. Acta Arith. 13, 177-236. Zbl 159.07101 Schlickewei H.P. (1990a): An upper bound for the number of subspaces occurring in the padic subspace theorem in Diophantine approximations. J. Reine Angew. Math. 406, 44-108 Schlickewei H.P. (1990b): An explicit upper bound for the number of solutions of the S-unit equations. J. Reine Angew. Math. 406, 109-120 Schlickewei H.P. (1992): The quantitative subspace theorem for number fields. Compos. Math. 82, 245-274 Schmidt W. (1970): Simultaneous approximation to algebraic numbers by ratio&s. Acta Math. 125, 189-201. Zbl 205.06702 Schmidt W. (1971a): Approximation to algebraic numbers. Enseign. Math., II. Ser. 17, 187-253. Zbl 226.10033 Schmidt W. (1971b): Linear forms with algebraic coefficients. J. Number Theory 3, 253-277. Zbl 221.10034 Schmidt W. (1971c): Linearformen mit algebraischen Koeffizienten. II. Math. Ann. 191, l-20. Zbl 207.35402 Schmidt W. (1972): Norm form equations. Ann. Math., II. Ser. 96, 526-551. Zbl 245.10008

References

335

Schmidt W. (1980): Diophantine Approximation. Lect. Notes Math. 785, Springer, Berlin Heidelberg New York. Zbl 421.10019 Schmidt W. (1989a): The number of solutions of norm form equations. Trans. Am. Math. Sot. 317, 197-227. Zbl 693.10014 Schmidt W. (1989b): The subspace theorem in Diophantine approximations. Compos. Math. 69, 121-173 Schmidt W. (1991): Diophantine Approximations and Diophantine Equations. Lect. Notes Math. 1467, Springer, Berlin Heidelberg New York. Zbl 794.11020 Schneider Th. (1934): Transzendenzuntersuchungen periodischer F’unktionen. J. Reine Angew. Math. 172, 65-69 and 70-74. Zbl 010.10501/010.10601 Schneider Th. (1936): Uber die Approximation algebraischer Zahlen. J. Reine Angew. Math. 175, 182-192. Zbl 014.20501 Schneider Th. (1937): Arithmetische Untersuchungen elliptischer Integrale. Math. Ann. 113, 1-13. Zbl 014.20402 Schneider Th. (1941): Zur Theorie der Abelschen F’unktionen und Integrale. J. Reine Angew. Math. 183, 110-128. Zbl 024.15504 Schneider Th. (1957): Einfiihrung in die transzendenten Zahlen. Springer, Berlin. Zbl 077.04703 Segal B.I. (1939): On integers with a certain form of prime factorization. Izv. Akad. Nauk SSSR, Ser. Mat. 3, 519-538. Zbl 024.29502 SegaI B.I. (1940): On certain sequences of integers. Izv. Akad. Nauk SSSR, Ser. Mat. 4, 319-334. Zbl 024.25003 Serre J.-P. (1989): Lectures on the Mordell-Weil Theorem. Vieweg, Braunschweig Shafarevich I.R. (1994): Basic Algebraic Geometry. Springer, Berlin Heidelberg New York. Zbl 797.14001 Shallit J. (1979): Simple continued fractions for some irrational numbers. J. Number Theory 11, 209-217. Zbl 404.10003 Shestakov S.O. (1991): On transcendence degree of some fields generated by values of elliptic function. Vestn. Mosk. Univ., Ser. I, 46, No. 3, 30-36. English transl.: Most. Univ. Math. Bull. 46, No. 3, 29-33. Zbl 735.11033 Shestakov S.O. (1992): On the measure of algebraic independence of certain numbers. Vestn. Mosk. Univ., Ser. I, 47, No. 2,8-12. English transl.: Most. Univ. Math. Bull. 47, No. 2, 8-11. Zbl 769.11030 Shestakova N.V. (1990): Estimates for reciprocal transcendence of certain numbers. Vestn. Mosk. Univ., Ser. I, 45, No. 6, 14-18. English transl.: Most. Univ. Math. Bull. 45, No. 6, 15-18. Zbl 722.11037 Shidlovskij A.B. (1954): On transcendence and algebraic independence of the values of certain classes of entire functions. Dokl. Akad. Nauk SSSR 96, 697-700. Zbl 055.27801 Shidlovskij A.B. (1955a): A criterion for algebraic independence of the values of a class of entire functions. Dokl. Akad. Nauk SSSR 100, 221-224. Zbl 064.04605 Shidlovskij A.B. (1955b): On certain classes of transcendental numbers. Dokl. Akad. Nauk SSSR 103, 977-980. Zbl 065.03301 Shidlovskij A.B. (1959a): A criterion for algebraic independence of the values of a .class of entire functions. Izv. Akad. Nauk SSSR, Ser. Mat. 23,35-66. ZblO85.27301 Shidlovskij A.B. (1959b): On transcendence and algebraic independence of the vaIues of certain classes of entire functions. Uch. Zap. Mosk. Univ. 186, 11-70. Zbl 126.07704 Shidlovskij A.B. (1959c): On transcendence and algebraic independence of the values of certain functions. Tr. Mosk. Mat. O.-va 8, 283-320, and corrigendum in ‘I%. Mosk. Mat. O.-va 10, 438. English transl.: Transl., II. Ser., Am. Math. Sot. 27, 191-230 (1963). Zbl 127.02302

336

References

Shidlovskij A.B. (1961): On a generalization of Lindemann’s theorem. Dokl. Akad. Nauk SSSR 138, 1301-1304. English transl.: Sov. Math., Dokl. 2, 841-844. Zbl 109.27602 Shidlovskij A.B. (1967a): On estimates for the transcendence measure of the values of E-functions. Usp. Mat. Nauk 2.2, No. 3, 245-246 Shidlovskij A.B. (1967b): Transcendence measure estimates for the values of Efunctions. Mat. Zametki 2, 33-44. English transl.: Math. Notes 2, 508-513. Zbl 173.05002 Shidlovskii A.B. (1979): On the estimates of the algebraic independence measures of the values of E-functions. J. Aust. Math. Sot. 27, 385-407. Zbl 405.10024 Shidlovskij A.B. (1981): On estimates for polynomials in values of E-functions. Mat. Sb. 115 (157), 3-34. English transl.: Math. USSR, Sb. 43, l-32. Zbl465.10027 Shidlovskij A.B. (1982): Diophantine Approximation and Transcendental Numbers. Izd. Mosk. Univ., Moscow. Zbl 517.10034 Shidlovskij A.B. (1987): Algebraic independence on the values of E-functions at algebraic points. Vestn. Mosk. Univ., Ser. I, 42, No. 1, 30-33. English transl.: Most. Univ. Math. Bull. 42, No. 1, 37-41. Zbl 624.10028 Shidlovskij A.B. (1989a): Transcendental Numbers. Walter de Gruyter, Berlin New York. Zbl 689.10043 Shidlovskij A.B. (1989b): Algebraic independence of values of E-functions at algebraic points. Mat. Zametki 45, 115-122. English transl.: Math. Notes 45, 78-83. Zbl 691.10028 Shidlovskij A.B. (1989c): Criterion of algebraic independence of the values of an Efunction at algebraic points. Vestn. Mosk. Univ., Ser. I, 44, No. 6, 17-21. English transl.: Most. Univ. Math. Bull. 44, No. 6, 20-25. Zbl 846.11042 Shidlovskij A.B. (1992): On the arithmetic properties of the values of E-functions. In: Analytic and Probabilistic Methods in Number Theory, Vol. 2. VSP, Utrecht, 23-29. Zbl 768.11023 Shiga H., Wolfart J. (1995): Criteria for complex multiplication and transcendence properties of automorphic functions. J. Reine Angew. Math. 463, l-25 Shiokawa I. (1985): Rational approximations to the values of certain hypergeometric functions. In: Number Theory and Combinatorics. World Science, Singapore, 353367. Zbl 614.10030 Shiokawa I. (1988): Rational approximations to the Rogers-Ramanujan continued fraction. Acta Arith. 50, 23-30. Zbl 655.10031 Shmelev A.A. (1968a): Concerning algebraic independence of some transcendental numbers. Mat. Zametki 3, 51-58. English transl.: Math. Notes 3, 31-35. Zbl 155.37902 Shmelev A.A. (1968b): On algebraic independence of some numbers. Mat. Zametki 4, 525-532. English transl.: Math. Notes 4, 805-809. Zbl 191.05502 Shmelev A.A. (1969): On algebraic independence of the values of certain E-functions. Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 103-112. Zbl 209.08403 Shmelev A.A. (1970): On approximating the roots of some transcendental equations. Mat. Zametki 7, 203-210. English transl.: Math. Notes 7, 122-126. Zbl 219.10037 Shmelev A.A. (1971): A.O. Gel’fond’s method in the theory of transcendental numbers. Mat. Zametki 10, 415-426. English transl.: Math. Notes 10, 672-678. Zbl 254.10028 Shmelev A.A. (1972a): On the question of the algebraic independence of algebraic powers of algebraic numbers. Mat. Zametki 11, 635-644. English transl.: Math. Notes 11, 387-392. Zbl 254.10029 Shmelev A.A. (1972b): Joint approximations of values of an exponential function at nonalgebraic points. Vestn. Mosk. Univ., Ser. I, 27, No. 4, 25-33. English transl.: Most. Univ. Math. Bull. 27, No. 4, 91-98. Zbl 243.10030

References

337

Shmelev A.A. (1974): A criterion for algebraic dependence of transcendental numbers. Mat. Zametki 16, 553-562. English transl.: Math. Notes 16, 921-926. Zbl 307.10034 Shmelev A.A. (1975a): Algebraic independence of exponents. Mat. Zametki 17, 407418. English transl.: Math. Notes 17, 236-243. Zbl 315.10027 Shmelev A.A. (1975b): Simultaneous approximations of exponential functions by polynomials in a given transcendental number. Ukr. Mat. Zh. 27, 555-563. English transl.: Ukr. Math. J. 27, 459-466. Zbl 312.10019 Shmelev A.A. (1976a): Algebraic independence of values of exponential and elliptic functions. Mat. Zametki 20, 195-202. English transl.: Math. Notes 20, 669-672. Zbl 336.10025 Shmelev A.A. (1976b): Simultaneous approximations of exponents by transcendental numbers of certain classes. Mat. Zametki 20,305-314. English transl.: Math. Notes 20, 731-736. Zbl 336.10024 Shmelev A.A. (1981): Algebraic independence of certain numbers connected with the exponential and elliptic functions. Ukr. Mat. Zh. 33, 277-282. English transl.: Ukr. Math. J. 33, 216-220. Zbl 461.10030 Shmelev A.A. (1985): Simultaneous approximation of the exponential by transcendental numbers of a special form. Ukr. Mat. Zh. 37, 232-237. English transl.: Ukr. Math. J. 37, 198-202. Zbl 568.10018 Shorey T.N. (1976): On linear forms in the logarithms of algebraic numbers. Acta Arith. 30, 27-42. Zbl 327.10033 Shorey T.N., van der Poorten A., Schinzel A., Tijdeman R. (1977): Applications of the Gelfond-Baker method in Diophantine equations. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, 59-78. Zbl 371.10015 Shorey T.N., Stewart C. (1983): On the Diophantine equation ax2t + bxty + cy2 = d and pure powers in recurrence sequences. Math. Stand. 52, 24-36. Zbl 503.10011 Shorey T.N., Tijdeman R. (1976): New applications of Diophantine approximations to Diophantine equations. Math. Stand. 39, 5-18. Zbl 341.10017 Shorey T.N., Tijdeman R. (1986): Exponential Diophantine Equations. Cambridge Univ. Press, Cambridge. Zbl 606.10011 Siegel C.L. (1921a): Approximation algebraischer Zahlen. Math. Z. 10, 173-213. JFM 48.0163.04 Siegel CL. (1921b): Uber Naherungswerte algebraischer Zahlen. Math. Ann. 84, 80-99. JFM 48.0164.01 Siegel CL. (1926): The integer solutions of the equation y2 = axn + bznel +. . + k. J. Lond. Math. Sot., II.Ser. 1, 66-68. JFM 52.0149.02 Siegel C.L. (1929/1930): Uber einige Anwendungen Diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys-Math. Kl., No. 1, l-70. JFM 56.0180.01 Siegel C.L. (1932): Uber die Perioden elliptischer Funktionen. J. Reine Angew. Math. 167, 62-69. Zbl 003.26501 Siegel CL. (1937): Die Gleichung ax” - by” = c. Math. Ann. 114, 57-68. Zbl 015.38902 Siegel C.L. (1949): Transcendental Numbers. Princeton Univ. Press, Princeton. Zbl 039.04402 Siegel C.L. (1966): Gesammelte Abhandlungen, Vol. 1. Springer, Berlin New York. Zbl 143.00101 Siegel C.L. (1969): Abschatzung von Einheiten. Nachr. Akad. Wiss. Giitt., II. Math.Phys. 8, 71-86. Zbl 186.36703 Skolem Th. (1938): Diophantische Gleichungen. Springer, Berlin. Zbl 018.29302

338

References

Sorokin V.N. (1985): On the irrationality of the values of hypergeometric functions. Mat. Sb. 127 (169), 245-258. English transl.: Math. USSR, Sb. 55, 243-257. Zbl 575.10029 Sorokin V.N. (1989): Linear independence of logarithms of certain rational numbers. Mat. Zametki 46, 74-79. English transl.: Math. Notes 46, 727-730. Zbl 694.10032 Sprindzhuk V.G. (1968): The irrationality of the values of certain transcendental functions. Izv. Akad. Nauk SSSR, Ser. Mat. 32, 93-107. English transl.: Math. USSR, Izv. 2, 89-104. Zbl 207.35403 Sprindzhuk V.G. (1979): Metric Theory of Diophantine Approximations. Wiley, New York. Zbl 482.10047 Sprindzhuk V.G. (1980): Achievements and problems in Diophantine approximation theory. Usp. Mat. Nauk 35, No. 4, 3-68. English transl.: Russ. Math. Surv. 35, No. 4, l-80. Zbl 438.10024 Sprindzhuk V.G. (1983): Arithmetic specializations of polynomials. J. Reine Angew. Math. 340, 26-52. Zbl 497.12001 Sprindzhuk V.G. (1993): Classical Diophantine Equations. Springer, Berlin Heidelberg New York. Zbl 787.11008 Stark H. (1967): There is no tenth complex quadratic field with class-number one. Proc. Natl. Acad. Sci. USA 57, 216-221. Zbl 148.27803 Stark H. (1969): On the “gap” in a theorem of Heegner. J. Number Theory 1, 16-27. Zbl 198.37702 Stark H. (1971): A transcendence theorem for class-number problems. I. Ann. Math., II. Ser. 94, 153-173. Zbl 229.12010 Stark H. (1972): A transcendence theorem for class-number problems. II. Ann. Math., II. Ser. 96, 174-209. Zbl 244.12004 Stark H. (1973a): Further advances in the theory of linear forms in logarithms. In: Diophantine Approximation and Its Applications. Academic Press, London New York, 255-293. Zbl 264.10023 Stark H. (1973b): Effective estimates of solutions of some Diophantine equations. Acta Arith. 24, 251-259. Zbl 272.10010 Stewart C.L. (1975): The greatest prime factor of on - b”. Acta Arith. 26, 427-433. Zbl 313.10007 Stewart CL. (1977a): On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. Proc. Lond. Math. Sot. 35, 425-447. Zbl 389.10014 Stewart CL. (1977b): Primitive divisors of Lucas and Lehmer numbers. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, 79-92. Zbl 366.12002 Stewart C.L. (1980): On divisors of terms of linear recurrence sequences. J. Reine Angew. Math. 333, 12-31. Zbl 475.10009 Stewart C.L. (1985): On the greatest prime factor of terms of a linear recurrence sequence. Rocky Mt. J. Math. 15, 599-608. Zbl 583.10006 Stewart C.L., Yu Kun-rui (1991): On the abc conjecture. Math. Ann. 291, 225-230. Zbl 761.11030 Stieltjes T. (1894/1895): Recherches sur les fractions continues. Ann. Fat. Sci. .Toulouse 8, 5.1-122 and A.l-47 (Reprint: ibid. VI. Ser., Math. 4, No. 1, J.i-iv, 5.1-35 (1995)) Stihl Th. (1984): Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann. 268, 21-41. Zbl 533.10031 Stolarsky K.B. (1974): Algebraic Numbers and Diophantine Approximations. M. Dekker, New York. Zbl 285.10022 Tamura J. (1991): Symmetric continued fractions related to certain series. J. Number Theory 38, 251-264. Zbl 734.11005 Tamura J. (1992): A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm. Acta Arith. 61, 51-67. Zbl 747.11029

References

339

Tanaka T. (1996): Algebraic independence of the values of power series generated by linear recurrences. Acta Arith. 74, 177-190 Thue A. (1908a): Bemerkungen iiber gewisse Naerungsbriiche algebraischer Zahlen. Norske Vid. Selsk. Skr. 3, l-34. JFM 39.0240.11 Thue A. (1908b): Om en generel i store hele tai ulosbar ligning. Norske Vid. Selsk. Skr. 7, 1-15. JFM 40.0256.01 Thue A. (1909): Uber Annaherungswerte algebraischer Zahlen. J. Reine Angew. Math. 135, 284-305. JFM 40.0265.01 Thue A. (1917): Uber die Unliisbarkeit der Gleichung ax2 + bx + c = dy” in grossen ganzen Zahlen x und y. Arch. Math. Naturvid. 34, l-6. JFM 46.0215.02 Thue A. (1918): Berechnung aher LSsungen gewisser Gleichungen von der Form ’ - by? = f. Norske Vid. Selsk. Skr. 2, No. 4, l-9. JFM 46.1448.07 Th”u”e A. (1977): Selected Mathematical Papers. Universitetsfdrlaget, Oslo. Zbl 359.01021 Tijdeman R. (1971a): On the number of zeros of general exponential polynomials. Indag. Math. 33, l-7. Zbl 211.09201 Tijdeman R. (1971b): On the algebraic independence of certain numbers. Indag. Math. 33, 146-162. Zbl 212.07103 Tijdeman R. (1972): On the maximal distance of numbers with a large prime factor. J. Lond. Math. Sot., II. Ser. 5, 313-320. Zbl 238.10018 Tijdeman R. (1973a): An auxiliary result in the theory of transcendental numbers. J. Number Theory 5, 80-94. Zbl 254.10027 Tijdeman R. (1973b): On integers with many small prime factors. Compos. Math. 26, 319-330. Zbl 267.10056 Tijdeman R. (1974/1976): Application of the Gelfond-Baker method to rational number theory. Colloq. Math. Sot. Janos Bolyai 13, 399-416. Zbl 335.10022 Tijdeman R. (1976a): Hilbert’s seventh problem. Proc. Symp. Pure Math. 28, 241268. Zbl 341.10026 Tijdeman R. (1976b): On the equation of Catalan. Acta Arith. 29, 197-209. Zbl 316.10008 Titenko V.V. (1987): On approximating the values of certain entire functions. Graduation Thesis, Mechanico-Mathematical Faculty, Moscow State University Tijpfer T. (1994): An axiomatization of Nesterenko’s method and applications on Mahler type functions. J. Number Theory 49, l-26. Zbl 813.11043 Toyoda M., Yasuda T. (1986): On the algebraic independence of certain numbers connected with the exponential and elliptic functions. Tokyo J. Math. 9, 29-40. Zbl 606.10030 TschakaIoff L. (1920): Arithmetische Eigenschaften der unendlichen Reihe ~~a-~.~~(“+~). I. Math. Ann. 80, 62-74. JFM 47.0167.02 c TschakaIoff L. (1921): Arithmetische Eigenschaften der unendlichen Reihe II. Math. Ann. 84, 100-114. JFM 48.0196.02 c x”a -“.5“(v+1). Tubbs R. (1983): A transcendence measure for some special values of elliptic functions. Proc. Am. Math. Sot. 88, 189-196. Zbl 517.10037 Tubbs R. (1986a): On the measure of algebraic independence of certain values of elliptic functions. J. Number Theory 23, 60-79. Zbl 591.10025 Tubbs R. (1986b): Independance algebrique et groupes algebriques. C.R. Acad. Sci., Paris, Ser. A 302, 91-94. Zbl 591.10026 Tubbs R. (1987a): Elliptic curves in two-dimensional abelian varieties and the algebraic independence of certain numbers. Mich. Math. J. 34, 173-182. Zbl632.10037 Tubbs R. (1987b): Algebraic groups and small transcendence degree. I. J. Number Theory 25, 279-307. Zbl 608.10036

340

References

Tubbs R. (1989): The algebraic independence of three numbers: Schneider’s method. In: ThCorie de Nombres, Quebec 1987. Walter de Gruyter, Berlin New York, 942_ 967. Zbl 678.10024 Tubbs R. (1990a): Algebraic groups and small transcendence degree. II. J. Number Theory 35, 109-127. Zbl 712.11039 Tubbs R. (1990b): A note on some elementary measures of algebraic independence. Proc. Am. Math. Sot. 109, 297-304. Zbl 702.11043 Tzanakis N. (1996): Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations. Acta Arith. 75, 165-190 Tzanakis N., de Weger B.M.M. (1989): On the practical solution of the Thue equation. J. Number Theory 31, 99-132. Zbl 657.10014 Tzanakis N., de Weger B.M.M. (1992): How to explicitly solve a Thue-Mahler equation. Compos. Math. 84, 223-288. Zbl 773.11023 Vlln%nen K. (1976a): On lower estimates for linear forms involving certain transcendental numbers. Bull. Aust. Math. Sot. 14, 161-179. Zbl 317.10043 VBBn%nen K. (1976b): On linear forms of the values of one class of E-functions. Acta Univ. Ouluen., Ser. A, No. 41, 1-19. Zbl 329.10025 Vatiinlnen K. (1977): On the simultaneous approximation of certain numbers. J. Reine Angew. Math. 296, 205-211. Zbl 356.10025 Vlaninlnen K. (1980): On linear forms of a certain class of G-functions and padic G-functions. Acta Arith. 36, 273-295. Zbl 429.10022 VB%nininen K., Wallisser R. (1989): Zu einem Satz von Skolem iiber die lineare Unabhangigkeit von Werten gewisser Thetareihen. Manuscr. Math. 65, 199-212. Zbl 685.10025 VBaninBnen K., Wallisser R. (1991): A linear independence measure for certain padic numbers. J. Number Theory 39, 225-236. Zbl 747.11030 Vasilenko O.N. (1985a): Arithmetic properties of values of polylogarithms. Vestn. Mosk. Univ., Ser. I, 40, No. 1, 42-45. English transl.: Most. Univ. Math. Bull. 40, No. 1, 63-68. Zbl 572.10026 Vasilenko O.N. (1985b): Irrationality of values of Gauss hypergeometric function. Vestn. Mosk. Univ., Ser. I, 40, No. 3, 15-18. English transl.: Most. Univ. Math. Bull. 40, No. 3, 20-24. Zbl 574.10037 Vasilenko O.N. (1985c): Linear independence of values of some hypergeometric functions. Vestn. Mosk. Univ., Ser. I, 40, No. 5, 34-37. English transl.: Most. Univ. Math. Bull. 40, No. 5, 40-43. Zbl 591.10024 Vasilenko O.N. (1985d): On approximation of hypergeometric functions and their Part I, Izd. Mosk. Univ., 10-16. Zbl values. Diophantine Approximations, 645.10033 Vasilenko O.N. (1986): On linear independence of the values of certain functions. Diophantine Approximations, Part II, Izd. Moskov. Univ., 3-12. Zbl 653.10032 Vasilenko O.N. (1991): On relation between approximations of exponents and approximations of $-functions. Vestn. Mosk. Univ., Ser. I, 46, No. 3, 57-58. English transl.: Most. Univ. Math. Bull. 46, No. 3, 42-43. Zbl 744.11035 Vasilenko O.N. (1993): On a construction of Diophantine approximations. Vestn. Mosk. Univ., Ser. I, 48, No. 5, 14-17. English transl.: Most. Univ. Math. Bull. 48, No. 5, 11-14. Zbl 848.11032 Viola C. (1985): On Dyson’s lemma. Ann. SC. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 105-135. Zbl 596.10032 Vojta P. (1991): Siegel’s theorem in the compact case. Ann. Math. 133, 509-548. Zbl 774.14019 van der Waerden B.L. (1958): Zur algebraischen Geometrie 19. Grundpolynom und zugeordnete Form. Math. Ann. 136, 139-155. Zbl 083.36703 Waldschmidt M. (1971): Indkpendance algebrique des valeurs de la fonction exponentielle. Bull. Sot. Math. Fr. 99, 285-304. Zbl 224.10033

References

341

Waldschmidt M. (1973a): Solution du huitieme probleme de Schneider. J. Number Theory 5, 191-202. Zbl 262.10021 Waldschmidt M. (1973b): Proprietes arithmetiques des valeurs de fonctions meromorphes algebriquement independantes. Acta Arith. 23, 19-88. Zbl 255.10039 Waldschmidt M. (1974a): Initiation aux nombres transcendants. Enseign. Math., II. Ser. 20, 53-85. Zbl 284.10012 Waldschmidt M. (197413): A propos de la methode de Baker. Bull. Sot. Math. Fr. Mem. 37, 181-192. Zbl 287.10021 Waldschmidt M. (1974c): Nombres Transcendants. Lect. Notes Math. 402, Springer, Berlin Heidelberg New York. Zbl 302.10030 Waldschmidt M. (1974/1975): Independance algebrique par la methode de G.V. Chudnovskij. Semin. D&urge-Pisot-Poitou, Theor. Nombres 16, 8/l-8/18. Zbl 325.10022 Waldschmidt M. (1975/1976): Suites colorees (d’apres G.V. Chudnovskij). Semin. Ddlange-Pisot-Poitou, Theor. Nombres 17, 21/l-21/11. Zbl 348.10024 Waldschmidt M. (1977a): Les travaux de G.V. Chudnovskij sur les nombres transcendants. Lect. Notes Math. 567, 274-292. Zbl 345.10020 Waldschmidt M. (1977b): On functions of several variables having algebraic Taylor coefficients. In: Transcendence Theory: Advances and Applications. Academic Press, London New York, 169-186. Zbl 366.10029 Waldschmidt M. (1977c): Proprietes arithmetiques de fonctions de plusieurs variables. II. Lect. Notes Math. 578, 108-133. Zbl 363.32003 Waldschmidt M. (1978a): Transcendence measures for exponentials and logarithms. J. Aust. Math. Sot. 25, 445-465. Zbl 388.10022 Waldschmidt M. (197813): Simultaneous approximation of numbers connected with the exponential function. J. Aust. Math. Sot. 25, 466-478. Zbl 388.10023 Waldschmidt M. (1979a): Nombres transcendants et fonctions sigma de Weierstrass. Math. Rep. Acad. Sci. Can. 1, 111-114. Zbl 402.10037 Waldschmidt M. (1979b): Transcendence methods. Queen’s Pap. Pure Appl. Math. 52, 132 p. Zbl 432.10016 Waldschmidt M. (1979c): Nombres transcendants et groupes algebriques. Asterisque 69-70, 1-218. Zbl 428.10017 Waldschmidt M. (1980): A lower bound for linear forms in logarithms. Acta Arith. 37, 257-283. Zbl 439.10020 Waldschmidt M. (1981): Transcendance et exponentielle en plusieurs variables. Invent. Math. 63, 97-127. Zbl 454.10020 Waldschmidt M. (1984a): Algebraic independence of values of exponential and elliptic functions. J. Indian Math. Sot., New Ser. 48, 215-228. Zbl 616.10029 Waldschmidt M. (1984b): Algebraic independence of transcendental numbers. Gelfond’s method and its developments. In: Perspectives in Mathematics. Birkhauser, Base1 Boston, 551-571. Zbl 556.10023 Waldschmidt M. (1984c): Independance algebrique et exponentielles en plusieurs variables. Lect. Notes Math. 1068, 268-279. Zbl 541.10031 Waldschmidt M. (1985): Petits degres de transcendance par la methode de Schneider en une variable. C.R. Math. Acad. Sci., Sot. R. Can. 7, 143-148. Zbl 571.10034 Waldschmidt M. (1986): Groupes algebriques et grandes degres de transcendance. Acta Math. 156, 253-302. Zbl 592.10028 Waldschmidt M. (1986/1987): Lemme de zeros pour les polynomes exponentiels. Semin. Arith. St.-fitienne Waldschmidt M. (1988): On the transcendence methods of Gelfond and Schneider in several variables. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 375-398. Zbl 659.10035

342

References

Waldschmidt M. (1991a): Nouvelles methodes pour minorer des combinaisons lineaires de logarithmes de nombres algkbriques. I. SCmin. ThCor. Nombres Bordx, Ser. II 3, 129-185. Zbl 733.11020 Waldschmidt M. (1991b): Nouvelles methodes pour minorer des combinaisons lineaires de logarithmes de nombres algebriques. II. Publ. Math. Univ. Paris-VI 93, No. 8 Waldschmidt M. (1991c): Fonctions auxiliaires et fonctionnelles analytiques. J. Anal. Math. 56, 231-279. Zbl 742.11035 Waldschmidt M. (1992a): Constructions de fonctions auxiliaires. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1990. Walter de Gruyter, Berlin New York, 285-307. Zbl 770.11038 Waldschmidt M. (1992b): Linear independence of logarithms of algebraic numbers. Matscience Lect. Notes, Madras, 168 p. Zbl 809.11038 Waldschmidt M. (1993): Minorations de combinaisons IinCaires de logarithmes de nombres algebriques. Can. J. Math. 45, 176-224. Zbl 774.11036 Waldschmidt M., Zhu Yao Chen (1983): Une generahsation en plusieurs variables d’un critere de transcendance de Gel’fond. C.R. Acad. Sci., Paris, Ser. A 297, 229-232. Zbl 531.10037 Walhsser R. (1971): Zur Transzendenz von Zahlen der Form nyEL=,ofi. Ber. Math. Forschungsinst. Oberwolfach 5, 211-222. Zbl 223.10020 Wahisser R. (1973a): Uber Produkte transzendenter Zahlen. J. Reine Angew. Math. 258, 62-78. Zbl 257.10014 WaIlisser R. (1973b): Uber die arithmetische Natur der Werte der Losungen einer Funktionalgleichung von H. Poincare. Acta Arith. 25, 81-92. Zbl 283.10019 Wallisser R. (1985): Rationale Approximation des q-Analogs der Exponentialfunktion und Irrationahtatsaussagen fiir diese Funktion. Arch. Mat. 44, 59-64. Zbl 558.33004 de Weger B.M.M. (1989): Algorithms for Diophantine equations. CWI Tract No. 65, Centre for Math. and Comp. Sci., Amsterdam. Zbl 687.10013 Weierstrass K. (1885): Zu Lindemann’s Abhandlung: Uber die Ludolph’sche Zahl. Sitzungsber. Preuss. Akad. Wiss., 1067-1085. JFM 17.0414.01 Wiles A. (1995): Modular elliptic curves and Fermat’s Last Theorem. Ann. Math., II. Ser. 141, 443-551 Wirsing E.A. (1961): Approximation mit algebraischen Zahlen beschrankten Grades. J. Reine Angew. Math. 206, 67-77. Zbl 097.03503 Wolfart J. (1981): Transzendente Zahlen aIs Fourierkoeffizienten von Heckes Modulformen. Acta Arith. 39, 193-205. Zbl 464.10015 Wolfart J. (1983): Eine arithmetische Eigenschaft automorpher Formen zu gewissen nichtarithmetischen Gruppen. Math. Ann. 262, 1-21. Zbl 499.10028 Wolfart J. (1984): Taylorentwicklungen automorpher Formen und ein Transzendenzproblem aus der Uniformisierungstheorie. Abh. Math. Semin. Univ. Hamburg 54, 25-33. Zbl 556.10020 Wolfart J. (1985/1986): Fonctions hypergeometriques, arguments exceptionels et groupes de monodromie. Publ. Math. Univ. Paris-VI 79, No. 9 Wolfart J. (1987): Values of Gauss’ continued fractions. Colloq. Math. Sot. Janos Bolyai 51, 1051-1063. Zbl 704.11017 Wolfart J. (1988): Werte hypergeometrischer Funktionen. Invent. Math. 92, 187-216. Zbl 649.10022 Wolfart J., Wiistholz G (1985): Der Uberlagerungradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen. Math. Ann. 273, 1-15. Zbl 559.14023 Wiistholz G. (1977): Zum Franklin-Schneiderschen Satz. Manuscr. Math. 20, 335354. Zbl 352.10015

References

343

Wiistholz G. (1978a): Linearformen in Logarithmen von U-Zahlen mit ganzzahligen Koeffizienten. J. Reine Angew. Math. 299/3&l, 138-150. Zbl367.10031 Wiistholz G. (1978b): Linearformen in Logarithmen von U-Zahlen und Transzendenz von Potenzen. I. Arch. Math. 30, 138-142. Zbl 396.10019 Wiistholz G. (1979): Linearformen in Logarithmen von U-Zahlen und Transzendenz von Potenzen. II. Arch. Math. 32, 356-367. Zbl 411.10008 Wiistholz G. (1980): Algebraische Unabhangigkeit von Werten von Funktionen, die gewissen Differentialgleichungen genfigen. J. Reine Angew. Math. 317, 102-119. Zbl 422.10028 Wiistholz G. (1982): Sur l’analogue abelien du theoreme de Lindemann. C.R. Acad. Sci., Paris, Ser. A ,295, 35-37. Zbl 498.10021 Wfistholz G. (1983a): Some remarks on a conjecture of Waldschmidt. In: Approx. Diophant. Nombres Transcendants, Colloq. Luminy 1982. Birkhauser, Base1 Boston, 329-336. Zbl 534.10026 Wiistholz G. (1983b): Uber das Abelsche AnaIogon des Lindemannschen Satzes. I. Invent. Math. 72, 363-388. Zbl 528.10024 Wiistholz G. (1984a): Transzendenzeigenschaften von Perioden elliptischer Integrale. J. Reine Angew. Math. 354, 164-174. Zbl 543.10025 Wiistholz G. (1984b): Zum Periodenproblem. Invent. Math. 78, 381-391. Zbl 548.10022 Wiistholz G. (1984c): Recent progress in transcendence theory. Lect. Notes Math. 1068, 280-296. Zbl 543.10024 Wiistholz G. (1987a): A new approach to Baker’s theorem on linear forms. I. Lect. Notes Math. 1290, 189-202. Zbl 632.10033 Wiistholz G. (1987b): ‘A new approach to Baker’s theorem on linear forms. II. Lect. Notes Math. 1290, 203-211. Zbl 632.10033 Wiistholz G. (1988): A new approach to Baker’s theorem on linear forms. III. In: New Advances in Transcendence Theory. Cambridge Univ. Press, Cambridge, 399-410. Zbl 659.10036 Wiistholz G. (1989a): Multiplicity estimates on group varieties. Ann. Math. 129, 471-500. Zbl 675.10024 Wiistholz G. (1989b): Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen. Ann. Math. 129, 501-517. Zbl 675.10025 Yakabe I. (1959): Note on the arithmetic properties of Kummer’s function. Mem. Fat. Sci., Kyushu Univ., Ser. A 13, 49-52. Zbl 090.26202 Yu K. (1985): Linear forms in elliptic logarithms. J. Number Theory 20, l-69. Zbl 555.10017 Yu K. (1989): Linear forms in padic logarithms. I. Acta Arith. 53, 107-186. Zbl 699.10050 Yu K. (1990): Linear forms in padic logarithms. II. Compos. Math. 74, 15-113. Zbl 723.11034 Yushkevich A.P. (1948): Leibniz and the origins of caIculus. Usp. Mat. Nauk 3, 150-164. Zbl 031.14502 Zariski O., Samuel P. (1960): Commutative Algebra. Van Nostrand, Princeton London Toronto Melbourne. Zbl 121.27801 Zudilin B.B. (1995): On rational approximations of values of a class of entire functions. Mat. Sb. 186, 89-124

344

Index

Index Algebraic - dependence, - independence, - integer, 74 - number, 11 - unit, 74 Approximation - Diophantine, - of algebraic - rational, 23, -- of e, 112 - simultaneous, Bombieri’s

17 17

78 numbers, 26, 78

15, 110

78, 79

Theorem,

Confluent Function, Conjugate, 22 Continued Fraction, - and e”, 112 - for functions, 95 - irregular, 95 Convergent, 13, 24

51 97 13, 24, 78

Degree - of a polynomial, 22 - of an algebraic number, 22 Diophantine - approximation, 78 - equation, 15, 29 Dirichlet - pigeonhole principle, 40 - unit theorem, 74 Dual Basis, 71 Dyson’s - inequality, 49 - lemma, 50 Effectiveness, 33, 34, 36, 45, 78 Euclidean Algorithm, 13 Functional Approximation, - of Hermite-Pad& 98

93

Gauss - class number 1 problem, 16 - hypergeometric function, 95, 112 Gel’fond’s - method, 82 - theorem, 48 Geometry of Numbers, 70

Height - of a polynomial, 22 - of an algebraic number, Hermite’s - identity, 84, 92 - method, 80, 83 HermitePade Functional tion, 98 Hilbert’s - seventh problem, 100 Hypergeometric - function, 95, 112

22

Approxima-

Index of a Polynomial, 56 Irrationality - exponent, 106 - measure, 17 -- of rr, 106 - of ir, 13, 113 - of Jz, 11 - of e, 78 Irregular Continued Fraction, Jacobi Identity,

95

97

Length - of a polynomial, 22 - of an algebraic number, 22 Lindemann’s Theorem, 88 Lindemann-Weierstrass Theorem, 103 Linear Form, 39, 64 Liouville’s - numbers, 26, 27 - theorem, 26, 44 Logarithm of Algebraic Number - transcendence measure, 104 Mahler ‘s - theorem, 54, 76 Measure - of algebraic independence, 18 - of irrationality, 17 -- of rr, 106 - of linear independence, 17 - of relative transcendence, 18 - of transcendence, 18 -- of ?r, 104 -- of e, 99 Minimal Polynomial, 22

88,

Index Minkowski’s Theorem, 66 Module - full, 74 - in a number field, 74 - non-degenerate, 75 - similar, 75 Non-archimedean Metrics, 40, 76 Norm, 22 Norm Form Equation, 73 Pade Approximation, 95 - of first kind, 98 - of second kind, 98 Partial Quotient, 24 Pell’s Equation, 30 Pigeonhole Principle, 40 Quadratic

Irrationality,

Rational - approximation, -- of e, 112 - subspace, 68 Roth’s - lemma, 57 - theorem, 55 Schmidt’s

23, 26

25

345

- approximation theorem, - finiteness theorem, 75 - subspace theorem, 68 Schneider’s - theorem, 54 Siegel’s - method, 53, 82 - theorem, 45, 53 Squaring the Circle, 13 Successive Minima, 71 Thue’s - equation, 28, 29 - method, 31, 41, 45, 53 - theorem, 42 Transcendence - measure, 18 -- of 7r, 104 -- of e, 99 -- relative, 18 - of r, 13 - of e, 83 - of e*, 82, 100 Transcendental - number, 11, 14 Transfer Theorem, 70 Waring

Problem,

16

60, 64, 65