Model Theory of Fields (Lecture Notes in Logic)

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Lecture Notes in Logic

5

D. Marker MeMessmer A.Pillay

Theory of Fie

Springer

Editorial Policy

§1. Submissions are invited in the following categories: i) ii) iii) iv)

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Lecture Notes in Logic Editors: S. Buss, San Diego J.-Y. Girard, Marseille A. Lachlan, Burnaby T. Slaman, Chicago A. Urquhart, Toronto H. Woodin, Berkeley

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D. Marker M. Messmer A. Pillay

Model Theory of Fields

Springer

Authors David Marker Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 85 IS. Morgan St. (M/C 249) IL 60607-7045 Chicago, USA E-mail: [email protected] Margit Messmer 1808 E. Bader IN 46617 South Bend, USA E-mail: mmessmer%mathcs%[email protected] Anand Pillay Department of Mathematics Notre Dame University IN 46556 Notre Dame, USA E-mail: [email protected]

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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Model theory of fields / D. Marker M. Messmer A. Pillay. Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Tokyo : Springer, 1996 (Lecture notes in logic 5) ISBN 3-540-60741-2 NE: Marker, David; Messmer, Margit; Pillay, Anand; GT Mathematics Subject Classification (1991): 03C60 ISBN 3-540-60741-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany Typesetting: Camera-ready by the authors SPIN: 10134704 2146/3140-543210 - Printed on acid-free paper

Contents

Preface I. Introduction to the Model Theory of Fields by David Marker

1

II. Model Theory of Differential Fields by David Marker

38

III. Differential Algebraic Groups and the Number of Countable Differentially Closed Fields by Anand Pillay

114

IV. Some Model Theory of Separably Closed Fields by Mar git Messmer

135

Preface The model theory of fields is a fascinating subject stretching from Tarski's work on the decidability of the theories of the real and complex fields to Hrushovksi's recent proof of the Mordell-Lang conjecture for function fields. Our goal in this volume is to give an introduction to this fascinating area concentrating on connections to stability theory. The first paper Introduction to the model theory of fields begins by introducing the method of quantifier elimination and applying it to study the definable sets in algebraically closed fields and real closed fields. These first sections are aimed for beginning logic students and can easily be incorporated into a first graduate course in logic. They can also be easily read by mathematicians from other areas. Algebraically closed fields are an important examples of ω-stable theories. Indeed in section 5 we prove Macintyre's result that that any infinite cj-stable field is algebraically closed. The last section surveys some results on algebraically closed fields motivated by Zilber's conjecture on the nature of strongly minimal sets. These notes were originally prepared for a two week series of lecture scheduled to be given in Bejing in 1989. Because of the Tinnanmen square massacre these lectures were never given. The second paper Model theory of differential fields is based on a course given at the University of Illinois at Chicago in 1991. Differentially closed fields provide a fascinating example for many model theoretic phenomena (Sacks referred to differentially closed fields as the "least misleading example"). This paper begins with an introduction to the necessary differential algebra and elementary model theory of differential fields. Next we examine types, ranks and prime models, proving among other things that differential closures are not minimal and that for K > N 0 there are 2* non-isomorphic models. We conclude with a brief survey of differential Galois theory including Poizat's model theoretic proof of Kolchin's result that the differential Galois group of a strongly normal extension is an algebraic group over the constants and the Pillay-Sokolovic result that any superstable differential field has no proper strongly normal expansions. Most of this article can be read by a beginning graduate student in model theory. At some points a deeper knowledge of stability theory or algebraic geometry will be helpful. When this course was given in 1991 there was an annoying gap in our knowledge about the model theory of differentially closed fields. Shelah had proved Vaught's conjecture for ω-stable theories. Thus we knew that there were N either N 0 or 2 ° non-isomorphic countable differentially closed fields, but did N not know which. In 1993 Hrushovski and Sokolovic showed there are 2 °. The proof used the Hrushovski-Zilber work on Zariski geometries and Buium's work on abelian varieties and differential algebraic groups. This circle of ideas is also crucial to Hrushovski's proof of the Mordell-Lang conjecture for function fields. The third paper, Differential algebraic groups and the number of countable

VI

differentially closed fields, gives a proof that the number of countable models is 2 K ° which avoids the Zariski geometry machinery. The final paper, Some model theory of separably closed fields , is a survey of the model theory of separably closed fields. For primes p > 0 there are separably closed fields which are not algebraically closed. These are the only other known example of stable fields. Separably closed fields play an essential role in Hrushovski's proof of the Mordell-Lang conjecture. This paper is intended as a survey of the background information one needs for Hrushovski's paper. We would like to thank the following people who gave helpful comments on earlier drafts of some of these chapters: John Baldwin, Elisabeth Bouscaren, Zoe Chatzidakis, Lou van den Dries, Wai Yan Pong, Sean Scroll, Zeljko Sokolivic, Patrick Speissegger and Carol Wood.

Suggestions for Further Reading There are a number of important topics that we either barely touched on or omitted completely. We conclude by listing a few such topics and some suggested references. o-minimal theories: Many of the important geometric and structural properties of semialgebraic subsets of Rn generalize to arbitrary o-minimal structures. L. van den Dries, Tame topology and O-minimal structures, preprint. J. Knight, A. Pillay and C. Steinhorn, Definable sets in ordered structures II, Trans. AMS 295 (1986), 593-605. A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Trans. AMS 295 (1986), 565-592. One of the most exciting recent developments in logic is Wilkie's proof that the theory of the real field with exponentiation is model complete and o-minimal. Further o-minimal expansions can be built by adding bounded subanalytic sets. A. J. Wilkie, Some model completeness results for expansions of the ordered field of reals by Pfaffian functions and exponentiation, Journal AMS (to appear). J. Denef and L. van den Dries, p-adic and real subanalytic sets, Ann. Math. 128 (1988), 79-138. L. van den Dries, A. Macintyre, and D.Marker, The elementary theory of restricted analytic fields with exponentiation, Annals of Math. 140 (1994), 183205.

Vll

r>-adic fields: There is also a well developed model theory of the p-adic numbers beginning with the Ax-Kochen proof of Artin's conjecture and leading to Denefs proof of the rationality of p-adic Poincare series. Macintyre's paper is a survey of the area. S. Kochen, Model theory of local fields, Logic Colloquium '74, G. Mueller ed., Springer Lecture Notes in Mathematics 499, 1975, 384-425. J. Denef, The rationality of the Poincare series associated to the p-adic points on a variety, Inv. Math. 77 (1984), 1-23. A. Macintyre, Twenty years of p-adic model theory, Logic Colloquium '84, J Paris, A. Wilkie and G. Wilmers ed., North Holland 1986, 121-153. Pseudofinite fields: Ax showed that the theory of finite fields is decidable. This subject is carefully presented in Fried and Jarden's book. The Chatzidakis-van den Dries-Mac in tyre paper gives useful properties of definable sets. M. Fried and M. Jarden, Field Arithmetic, Springer-Verlag (1986). Z. Ghatzidakis, L. van den Dries and A. Macintyre, Definable sets over finite fields, J. reine angew. Math 427 (1992), 107-135. Zariski Geometries and the Mordell-Lang Conjecture: In 1992 Hrushovski gave a model theoretic proof of the Mordell-Lang conjecture for function fields. His work depends on a joint result with Zilber which characterizes the Zariski topology on an algebraic curve. The Bouscaren-Lascar volume is the proceedings of a conference in Manchester devoted to Hrushovski's proof and the model theoretic machinery needed in its proof. E. Hrushovski and B. Zilber, Zariski Geometries, Journal AMS (to appear). E. Hrushovski, The Mordell-Lang conjecture for function fields, Journal AMS (to appear). E. Bouscaren and D. Lascar, Stability Theory and Algebraic Geometry, an introduction preprint.

Introduction to the Model Theory of Fields David Marker University of Illinois, Chicago

My goal in these lectures is to survey some classical and recent results in model theoretic algebra. We will concentrate on the fields of real and complex numbers and discuss connections to pure model theory and algebraic geometry. Our basic language will be the language of rings Cτ = {+, —, ,0,1}. The field axioms, Tfieids, consists of the universal axioms for integral domains and the axiom VxBy (x — 0 V xy = 1). Since every integral domain can be extended to its fraction field, integral domains are exactly the /^-substructures of fields. For a fixed field F we will study the subsets of Fn which are defined in the language

§1 Algebraically closed fields Let ACF be Tfieids together with the axiom n-l n

Vα 0 . . . Vα n _ι3z x + ]Γ aixi = 0 t=0

for each n. Clearly ACF is not a complete theory since it does not decide the characteristic of the field. For each n let φn be the formula

Vz X + ... + S = 0. n times

For p prime, let ACFp be theory ACF + φp, and let ACFQ = ACF(J{->φn : n = 1,2,...}. For our purposes the key algebraic fact about algebraically closed fields is that they are described up to isomorphism by the characteristic and the transcendence degree. This has important model theoretic consequences. Recall that for a cardinal K a theory is K- categorical if there is, up to isomorphism, a unique model of cardinality K. Proposition 1.1. Let p be prime or zero and let /c be an uncountable cardinal. The theory ACFP is K- categorical, complete, and decidable.

Proof. The cardinality of an algebraically closed field of transcendence degree λ is equal to N0 + A. Thus the only algebraically closed field of characteristic p and cardinality « is the one of transcendence degree K. Vaught's test (a simple consequence of the Lόwenheim-Skolem theorem) asserts that if a theory is categorical in some infinite cardinal, then the theory is complete. Finally, any recursively axiomatized complete theory is decidable. Corollary 1.2. Let φ be an £r-sentence. Then the following are equivalent:

i)CM ϋ) ACFQ \= φ iii) ACFp \= φ for sufficiently large primes p. iv) ACFp \= φ for arbitrarily large primes p. Proof. Clearly ii)—> i), while i)—»ii) follows from the completeness of ACF$. If ACFo \= φ, then, since proofs are finite, there is an n such that ACF U {-i(^ι,..., -*φn} |= φ. Clearly if p> n is prime, then ACFP \= φ. Thus ii)—»iii). Clearly iii)—>iv) Suppose ACFQ £ φ. Then by completeness ACF0 |= -κ£, and by ii)—>iii), for sufficiently large primes p ACFP \= -ιφ. Thus there aren't arbitrarily large primes p where ACFP (= 0, so iv)—* ii). Corollary 1.2 has a surprising consequence. Theorem 1.3 (Ax [A]) Let / : C n -+ Cn be a polynomial map. If / is one to one, then / is onto. Proof. We can easily write down an £r-sentence Φ<j such that a field F \= Φ<j if and only if for any polynomial map / : Fn —* Fn where each coordinate function has degree at most rf, if / is one to one, then / is onto. By 1.2, it suffices to show that for sufficiently large primes p, ACFp \= Φd for all d £ N. Since ACFp is complete it suffices to show that if K is the algebraic closure of the p element field, then any one to one polynomial map / : Kn —> Kn is onto. If / : Kn -» Kn is a polynomial map, then there is a finite subfield KQ C K such that all coefficients in / come from KQ. Let x £ Kn. There is a finite KI C K such that K0 C KI and x G K%. Since / : K? -> K%, f is one to one and KI is finite, f\K\ must be onto. Thus x = f(y) for some y £ K%. So / is onto. This result was later given a completely geometric proof by Borel ( [B]). Definition. We say that an £-theory T has quantifier elimination if and only if for any £-formula φ(υι,..., υm) there is a quantifier free ^-formula ^ ( v i , . . . , vm) such that T \= Vv φ(v) <-»• ψ(v). The following theorem leads to an easy test for quantifier elimination.

Theorem 1.4. Let £ be a language containing at least one constant symbol. Let T be an £ theory and let φ(vι, . . . , vm) be an C, formula with free variables t>ι, . . , vm (we allow the possibility that ra = 0). The following are equivalent: i) There is a quantifier free ^-formula ψ(υι, . . , vm) such that T h Vϋ (φ(v) <-»•

tf(tθ)

ii) If A and B are models of T, C C .4 and C C β, then .4 |= φ(a) if and only if B \= φ(a) for all α G C. proof. [i) —»• ii)]: Suppose T h Vΰ (0(v) <->• Ψ(v)), where ^ is quantifier free. Let ά G C where C is a substructure of A and β and the later two structures are models of T. Since quantifier free formulas are preserved under substructure and extension A\=φ(a)~A^ψ(a) +->C\=ψ(a) (since C C A) ^B\=ψ(ά)

(since C C β)

[ii) -> i)]. First, if T h Vϋ <£(ϋ), then T \- Vv (φ(v) <-> c = c). Second, if T h Vϋ -^(t;), then T h Vv (<^(v) <-» c / c). In fact, if φ is not a sentence we could use "vi — vi" in place of c — c. Thus we may assume that both φ(v) and ~^φ(v) are consistent with T. Let Γ(v) = {φ(ϋ) : ψ is quantifier free and T h Vϋ (ψ(v) -* V'(v))}- Let di, . . . , dm be new constant symbols. We will show that T + Γ(d) h φ(d). Thus by compactness there are ψι, . . . , ψn G Γ such that T h Vi; (/\ψi(v) —> Φ(v)). So T h Vϊ; (^ ^t (v) <-*• φ(v)) and ^ VΊ'(^) is quantifier free. We need only prove the following claim. claim. Γ + Γ(J) h φ(d). Suppose not. Let A \= T + Γ(d) -h ~^φ(d). Let C be the substructure of A generated by d. [Note: if ra = 0 we need the constant symbol to insure C is non-empty.] Let Diag(C) be the set of all atomic and negated atomic formulas with parameters from C that are true in C. Let Σ = T + Diag(C) + φ(d). If Σ is inconsistent^ then there are quantifier free formulas quantifier free formulas V>ι(«0> >VVι(d) € Diag(C), such that T h Vv (/\Ψi(v -> ^Φ(v)). But then T h Vv (ψ(v) -»• V^i(v)). So\J^ψi(v) G Γ and C |= \/-*l>i(d), a contradiction. Thus Σ is consistent. Let β |= Σ. Since Σ D Diag(C), we may assume that C C B. But by a), since A (= ^(d), β f= - Ψ(d), a contradiction. The next lemma shows that to prove quantifier elimination for a theory we need only prove quantifier elimination for formulas of a very simple form.

Lemma 1.5. Suppose that for every quantifier free ^-formula θ(v,w), there is a quantifier free ψ(v) such that T h Vΰ (3w θ(v,w) <-»• V'(^))- Then every >C-formula (^(ΰ) is provably equivalent to a quantifier free £-formula. Proof. We prove this by induction on the complexity of φ. This is clear if φ(v) is quantifier free. For i = 0, 1 suppose that T h Vϋ (0, (ϋ) = ψi(v))) where ψt is quantifier free. If φ(v) = -.ffo(v), then T h Vv (/(i;) <-> --^o(t>)) If φ(v) = Θ0(v) Λ 0ι (ϋ), then T h Vv (<£(ΰ) <-> (ιfo(v) Λ VΊ(V))) In either case φ is provably equivalent to a quantifier free formula. Suppose that T h Vv(θ(v,w) «-+ ψ$(v,w}), where ^ is quantifier free. Suppose 0(ΰ) = 3ιt; ^(ϋ, w). Then T h Vϋ (<^(^) ^-> 3ty(^(v, w;)). By our assumptions there is a quantifier free ψ(v) such that T h Vϋ (3w; ΨQ(V,W) <-» ^(ϋ)). But then T h V ϋ (ψ(v) ^V ) (^)) Thus to show that T has quantifier elimination we need only verify that condition ii) of theorem 1.4 holds for every formula φ(v) of the form 3wθ(ϋ, w) where θ(v,w) is quantifier free. Theorem 1.6 The theory ACF has quantifier elimination. Proof. Let F be a field and let K and L be algebraically closed extensions of F. Suppose φ(v, w) is a quantifier free formula, a £ F, b £ K and K \= φ(b, a). We must show that L \= 3v φ(v,a). There are polynomials /ι,j,flf»j G F[X] such that φ(v,a) is equivalent to

Thenίf μ ΛΓ=ι /*j(*) = 0 Λ Λ?=1 ftjί*) for some i. Let F be the algebraic closure of F. We can view F as a subfield^pf both K and L. If any /,-j is not identically zero for j = 1, . . . , m, then b £ F C L and we are done. Otherwise since

»=ι

9ij(X) = 0 has finitely many solutions. Let {GI, . . . , cs} be all of the elements of L where some
Definition. A theory T is model complete if whenever M C N and M, N \= T, then N is an elementary extension of M. Since quantifier free formulas are preserved under substructure and extension, any theory with quantifier elimination is model complete. The model completeness of algebraically closed fields can also be proved be appealing to Lindstrom's result that any NI-categorical, VΞ-axiomatizable theory is model complete (see [C]). In fact, model completeness is a weak form of quantifier elimination. A theory T is model complete if and only if every formula is equivalent to one of the form Ξ v i , . . . , 3vnφ(v, w) where φ is quantifier free. For algebraically closed fields model completeness implies that if F C K are algebraically closed fields and Σ is a finite system of equations and inequations over F which have a solution in K, then Σ already has a solution in F. Model completeness gives a very simple proof of Hubert's Nullstellensatz. (We refer the reader to Lang ( [LI]) for all algebraic results. If F is a field and I C F[Xι,... ,Xn] is an ideal, let VF(I) = {a G Fn : /(α) = 0 for all / G /}. Corollary 1.7. (Nullstellensatz) If F is an algebraically closed field and P C F[Xι,..., Xn] is a prime ideal then VF(P) ^ 0. Proof. Let K be the algebraic closure of F[Xι,... ,Xn]/P. By model completeness K is an elementary extension of F. By Hubert's basis theorem, P is finitely generated. Say P = {/i,..., fm).— The sentence n

3vι...3υn /\/<(*!,...,v n ) = 0 «=ι is true in K, as (Xι/P,..., Xn/P) is a witness. By model completeness this sentence is true in F. Using the fact that y/Ί is a finite intersection of prime ideals, the above proof can easily be modified to show that if / is an ideal in F[X] and 1 ^ vT, then V>(/) φ 0. While model completeness is useful in some applications, quantifier elimination is the primary tool for understanding definable sets in algebraically closed fields. Definition. A theory T is strongly minimal if for any M \= T, every definable subset of M is either finite or cofinite. (Note that "definable" means "definable with parameters".) If F is algebraically closed, then every definable subset of F is a finite Boolean combination of sets of the form {x : f ( x ) = 0} where f ( X ) G F[X]. If f ( X ) is not identically zero, then the set of zeros of / is finite. Thus algebraically closed fields are strongly minimal.

Quantifier elimination also shows that the definable sets are exactly the constructible sets of algebraic geometry. n

Definition. If F is a field, we say that X C F is Zariski closed if it is a finite union of sets of the form m

{ * : / \ Λ ( * ) = 0} where /ι,...,/ m £ F[Xι,...,Xm]. By Hubert's basis theorem the intersection of a (possibly infinite) collection of Zariski closed sets is Zariski closed. Thus the Zariski closed sets give a topology on Fn. A subset of Fn is called constructible if it is a finite Boolean combination of Zariski closed sets. By quantifier elimination, if F is an algebraically closed field, then the definable sets are exactly the constructible ones. The following theorem of Chevalley gives the geometric restatement of quantifier elimination. Corollary 1.8. The projection of a constructible set is constructible. Definition. A Zariski closed set is irreducible if it can not be written as a union of two proper closed subsets. We will refer to irreducible closed sets as varieties. Since F[X] is Noetherian, there are no infinite descending chains of Zariski closed sets. Thus every Zariski closed set is a finite union of irreducible closed sets. Thus by quantifier elimination, if X is definable then X = UΓ=ι(^« ^ Φ) where Vi is an irreducible component of the Zariski closure of X and O, is Zariski open. Later we will give a description of the definable functions. Definition. If A is a commutative ring, let Spec(A) be the set of prime ideals of A. We call Spec(A) the Zariski spectrum of A. We topologize Spec(A) by taking basic closed sets {P : aι,..., an £ P} for α 1 } . . . , an £ A. The Zarsiki spectrum has a model theoretic analog. Definition. If T is a complete theory and M [= T, an n-type over M is a maximal set of formulas with parameters from M and free variables v\,..., vn that is consistent with T. Let Sn(M) be the set of n-types. We call Sn(M) the Stone Space of M. We topologize 5n(M) by taking basic open sets{p £ 5n(M) : φ £ p} for each formula φ with parameters from M. Note that these basic sets are indeed clop en. The compactness theorem for first order logic implies that 5n(M) is a compact space.

If F is an algebraically closed field there is a natural bijection between Sn(F) and Spec(F[Xll...ίXn]). If p is an n-type, let Ip = {f G F[X] : "/(vi, , vn) = 0" G p} It is easy to see that Ip is a prime ideal. Moreover, if I is any prime ideal, let p be the set of consequences of

By quantifier elimination, p G Sn(F). The map p H-> /p is easily seen to be continuous. Thus 5pec(F[X]) is compact. Definition. A complete theory T is u -stable if for any F [= T, 15^(^)1 = \F\. By Hubert's basis theorem all prime ideals are finitely generated. Thus |5pec(F[X])| = \F\ for any algebraically closed field F. By the above remarks |5n(F)| - \F\. Thus for p a prime or zero, ACFp is u -stable. Indeed a basic result from model theory says that NI -categorical theories are always u -stable. In u -stable theories there is a notion of Morley Rank which associates an ordinal to each definable set. In strongly minimal theories this notion is particular simple. Definition. Let M \= T (an arbitrary theory). Let α,6ι,...,6 n G M. We say that a is algebraic over 6 if there is an £-formula φ(v, w\, . . . , wn) such that M \= φ(a,b) and {x G M : M |= φ(x,b)} is finite. If T is strongly minimal then algebraic dependence satisfies the exchange lemma, namely if α is algebraic over 6, c and not algebraic over 6, then c is algebraic over 6,α. In algebraically closed fields this is exactly the usual notion of algebraic dependence. One can give a well defined notion of dimension, namely dim (ai, . . . , an) is the maximal cardinality of a subset {α^ , . . . , α,m } such that no α,^ is algebraic n over {a^ , . . . , α,m } \ {α^ }. If M \= T and X C M is definable, then the Morley rank of X is the maximum dimension of a tuple (61, . . . , 6n) such that for some elementary extension N of M N \=bζX. If X has Morley rank m, then the Morley degree of X is the maximum number of pairwise disjoint definable rank m sets X can be partioned into. Morley rank and degree have geometric meaning. Definition. If V is a Zariski closed set in Fn, let F[V] be the ring F[Xι,...,Xn]/I(V), where I(V) is the ideal of all polynomials which vanish at each point in V. We call F[V] the coordinate ring of V. If V is irreducible, then F[V] is an integral domain and we let F(V) be the fraction field of F[V]. We call F(V) the function field of V. The ring F[V] corresponds to the ring of polynomial functions on V, while F(V) corresponds to the field of (partial) rational functions on V. There is a classical dimension theory for varieties.

Definition. If V is an irreducible variety, we define the dimension of V to be the transcendence degree of F(V) over F. If X is a construetible set its dimension defined to be the maximal dimension of an irreducible component of the Zariski closure. Note that if O is an open subset of an irreducible variety V, then V \ O has dimension less than the dimension of V. Proposition 1.9. If V is a variety, then its Morley rank is equal to its dimension. Proof. If V is a variety of dimension m, then F(V) has transcendence degree m over F. Let K be the algebraic closure of F(V). Since Xι/I(V),.. .,Xn/I(V) generate F(V) over F they have transcendence degree m over F. Thus (Xι/I(V),... ,Xn/I(V)) demonstrates that V has Morely rank at least m. On the other hand, if L is a field extension of F and L |= ά G F, there is a ring homomorphsim from F[V] into L given by / »-*• /(α). Clearly the transcendence degree of ά is at most the transcendence degree of F[V] over F. Corollary 1.10. If X is a non-empty constructive set, then its Morley rank is equal to its dimension.

Proof. First suppose that V is an irreducible variety, O is open, and X = V Π O is non-empty. If p is the type such that V\ = V(Ip), then p is the type of maximal Morely rank in V. The type p must contain the formula "ϋ G O", as otherwise there is a polynomial / ^ Ip such that "f(v) — 0" G p, a contradiction. If X is an arbitrary cons true tible set, then X = (JfLi ^* ^ ^»> wnere V i , . , Vm are the irreducible components of the Zariski closures of X, O, is open, and V^ Π O, is non-empty. The corollary now follows from the first case. Finally, we will give the promised description of definable functions. Theorem 1.11. Let F be an algebraically closed field. Let / : Fn -> F be a definable function. Then there is a nonempty open set O such that: i) If F has characteristic 0, then there is a rational function r such that f\0 = r. ii) If F has characteristic p > 0, then there is a natural number n and a rational function r such that f\O = σ~n or, where σ is the Frobenious automorphism σ(x) = xp.

proof. Let K be an elementary extension of F containing <ι,...,ί n which are algebraically independent over F. Since f ( t ) are fixed by any automorphism of F which fixes t\t... ,ίn and F, f ( t ) is in the perfect closure of F(tι,... >tn). Thus in characteristic 0 there is a rational function r such that r(t) — f(t). In characteristic p > 0, we can find a rational function r and a natural number n such that σ-n(r(i)) = /(i).

Henceforth we consider only the characteristic zero case as the characteristic n p case is analogous. In F consider Y = {x 6 F : r(x) = t(x)}. Since r(ΐ) = f ( t ) and the ti are independent, Y has Morely rank n. Since there is a unique n-type of Morley rank n, Y has Morely rank n and -Y has Morely rank less than n. n Thus if V is the Zariski closure of -.y, dim V < n. Let O = F \ V. Then O is n a nonempty open subset of F and f\O = r. In [Pi4] Pillay provides a more extensive introduction to the model theory of algebraically closed fields.

§2 Real Closed Fields

We next turn our attention to the field of real numbers. We would like to prove model completeness and quantifier elimination results analogous to those for algebraically closed fields. There is one major difficulty: we can not eliminate quantifiers in the language of rings. In particular in the reals we can define the ordering by x < y & 3z (z2 + x = y Λ z ± 0) and we will see that that this is not equivalent to a quantifier free formula (in fact by a theorem of Macintyre, McKenna, and van den Dries ([M-M-D]). We circumvent this difficulty by extending £Γ to £OΓ = £r U {<}• In this language we will prove quantifier elimination. We begin by examining the work of Artin and Schrier on the algebraic structure of the real field (see [LI] for details). For the remainder of this section we all fields will have characteristic zero. The model theoretic study of the R began with the work of Tarski. See [D2] for further discussion of Tarski's work. Definition. A field F is said to be formally real if — 1 is not a sum of squares. We say F is real closed if it is formally real and has no proper formally real algebraic extensions. Lemma 2.1. If F is formally real, and a £ F is not a sum of squares, then F(y/^a) is formally real. It follows from 2.1, then if F is real closed and a φ 0, then exactly one of α and — α has a square root in F. One can then define an order on F such that the positive elements are exactly the squares. Clearly this is the only way to order F. Theorem 2.1. (Artin-Schrier) Let (F,<) be an ordered field. Then the following are equivalent.

10

i) F is real closed. ii) F(i) is algebraically closed (were i = \J— 1). iii) If p(X) G F[X], and α, 6 G F such that α < 6 and p(α) < p(6) then there is c £ F such that α < c < 6 and p(c) = 0. iv) For any α £ F either α or — α is a square and every polynomial of odd degree has a root. Since iv) does not mention the ordering, we can axiomatize the theory of real closed fields in the language Cτ by axioms asserting that F is formally real field of characteristic zero where iv) holds. We call this theory RCF. Definition. If F is formally real we say that K D F is a real closure of F if it is a real closed algebraic extension of F. Clearly every real field has a real closure, however, unlike algebraic closures, the real closure of a formally real field need not be unique. For example, if t is transcendental, Q(Vt) and Q(\/—?) are real. Let FI and F2 be real closures of Q(V?) and Q(^/~t) respectively. Both FI and F2 are real closures of Q(<), but they are not isomorphic over Q(ί). (Note that this shows that the theory RCF does not eliminate quantifiers in the language £Γ.) On the other hand, if (F, <) is an ordered field, then there is a unique real closure /£, where the ordering on K extend the ordering on F. The proof uses Sturm's algorithm to bound the location of the roots of a polynomial (see [LI]). Let RCOF be the theory of real closed ordered fields in the language £or. The axioms for RCF are the axioms for ordered fields and an axiom schema asserting the intermediate value theorem for polynomials (2.2 iii). Theorem 2.3. The theory RCOF has quantifier elimination in £OΓ. Proof. We apply theorem 1.4. Let F0 and FI be models of RCOF and let (R, <) be a common substructure. Then (Λ, <) is an ordered domain. Let L be the real closure of the fraction field of R. By the uniqueness of real closures we can may assume that (L, <) is a substructure of F0 and ί\. Suppose φ(υ, w) is quantifier free, ά E R, b G F0 and F0 (= φ(b,ά). We need to show that FI \= 3v φ(v,a). It suffices to show that L [= 3v φ(v^a). As in the proof of theorem 1.6 (and fooling around with the order), we may assume that there are polynomials /i , . . . , fn , gι , . . . , gm G R[X] such that φ(v,a) is

Λ /.» =

t=l

11 If any of the /; is not zero, then since φ(b, α), α is algebraic over R and thus in L. So we may assume φ(v,a) is

Since L is a real closed field, by 2.1 ii) we can factor each
/ \J ai
Since F0 |= 0(6, ά), for some t, α< < 6 < ft. Then L |= α). Corollary 2.4. RCOF and flCF are complete, model and decidable. proof Model completeness for RCOF is immediate from quantifier elimination. For RCF model completeness follows because if F C K are real closed fields, then, when viewed as £or-structures F is still a substructure of K. Thus K is an elementary extension in £or and hence in £Γ. Any real closed field contains (Q, <). Thus the real closure of Q, the real algebraic numbers, is an elementary submodel of every real closed field, so the theory is complete. Since RCF and RCOF are recursively axiomatized and complete, both are decidable. Since RCF and RCOF have the same models, we will forget about RCOF and refer to the theory as RCF. The next concept is the correct analog of strong minimality for ordered structures. Definition. A structure (M, <, . . .) is o-minimal if every definable subset of M is a finite union of points and intervals. Corollary 2.5. Every real closed field F is o-minimal. Proof. For f ( X ) £ -F, {x : f ( x ) > 0} is a union of intervals. From this o-minimality follows easily. The notion of o-minimality was introduced by van den Dries [Dl] and studied extensively by Pillay and Steinhorn, among others (see for example [PS] and [K-P-S]). Of particular interest is the fact that o-minimality leads to

12 a deep structure theory for definable sets in n-space. In §3 will give classical proofs of some of the consequences of o-minimality for real closed fields. Quantifier elimination leads to a geometric characterization of the definable sets. Let F be a real closed field Definition. We say that X C Fn is semialgebraic if it is a finite Boolean combination of sets of the form {x : f ( x ) > 0} or {x : f ( x ) = 0}, / € F[X]. Clearly, the semialgebraic sets are exactly the quantifier free definable sets. Quantifier elimination then has the following geometric interpretation. Corollary 2.6. (Tarski-Seidenberg Theorem) The projection of a semialgebraic set is semialgebraic. The next corollaries are typical applications. Corollary 2.7. If A is a semialgebraic set then the closure of A is semialgebraic. Proof. Let d(x, y) = z if and only if z1 — Σ(x* ~~ 2/*)2 anc^ z > O Then the closure of A is {x : Ve > 0 By y G A Λ d(x, y) < e}. Corollary 2.8. Let F be real closed. If X C Fn is a closed and bounded semialgebraic set and / : X —> Fm is continuous and semialgebraic, then the image of X is closed and bounded. Proof. If F = R this is trivial as X is compact if and only if X is closed and bounded and the continuous image of a compact set is compact. On the other hand if φ(v,a) defines X and ^(x^y^ϊ)) defines /. There is an £OΓ sentence Φ asserting that for all ά and β if φ(v, ά) is a closed bounded set Y and ψ(x, y, β) defines a continuous function with domain Y, then the image is closed and bounded. This sentence is true in R and hence true in F. Model completeness has several important applications. The first is Robinson's version of Artin's solution to Hubert's 17th problem. Definition. Let f(X\,... ,Xn) be a rational function over a real closed field R. We say that / is positive semi-definite if /(ά) > 0 for all α £ R. Theorem 2.9. (Artin) If / is a positive semi-definite rational function over a real closed field β, then / is a sum squares of rational functions over R. The proof uses one algebraic lemma (see [LI]).

13 Lemma 2.10. If F is real and a £ F is not a sum of squares, then there is an ordering of F where α is negative. Proof of 2.9.

Suppose f(Xι, . . . , Xn] is a positive semi-definite rational function which is not a sum of squares. Then, by 2.10, there is < an ordering of R(X) where / is negative. Let K be the real closure of the ordered field (R(X), <). Then K \= 3ϋ f ( v ) < 0. By model completeness this sentence also holds in 72, contradicting the fact that / is positive semi-definite. A similar argument can be used to prove the following real nullstenllensatz. Theorem 2.11. (Dubois-Reisler) Let R be a real closed field and let / be an ideal in R[X]. Then / = I(V(I)) if and only if αi, . . . , an G / whenever £ α? G /. (For a proof see [Di] or [B-C-R]). This style of argument can also be used (and seems essential) to prove some of the basic properties of Nash functions. We next examine the definable functions in real closed fields. We let R be a real closed field. Lemma 2.12. If / : R —» R is definable, then for any open set U C β, there is a point x G U such that / is continuous at x. Proof, (van den Dries [Dl]) By completeness it suffices to prove this for R. case 1: There is an open set V C U such that / has finite range on V. In this case we can find an open subset of V on which / is constant. case 2. Otherwise. We build VQ D V\ D open subsets of U such that the closure of Vn+\ is contained in Vn. Given Vn, let X be the range of / o n V^. By o- minimality X contains an interval (α, 6) of length less than £. Let Vn+ι be a suitable open subinterval of Vn Π /~ 1 (α, 6). Let x £ ΠK' Clearly / is continuous at x. n

n

Lemma 2.12 will generalize to R once we know that R can not be partitioned into finitely many sets with non-empty interior. Corollary 2.13. If / : R -* R is definable, then we can partition R = Iι U . . . In \JF where F is finite and the Ij are disjoint open sets where / is continuous on each Ij . Proof.

Otherwise, by o-minimality, {x : f is discontinuous at x} has non-empty interior, contradicting lemma 2.12.

14 m

n

Proposition 2.14 (van den Dries [D3]) Let X C R + be definable. There is a m n n definable function / : R -> R such that for all x G R™ if 3y G Λ (z, y) G ^, then (x, /(#)) G A'. (We say that the theory of real closed fields has definable Skolem functions.) Proof. By induction it suffices to prove this for n = 1. For α G R™ let Xa = {y : (α,y) G X}. By o-minimality Xa is a finite union of points and intervals. If Xa is empty let /(α) = 0, otherwise we define /(α) by cases. case case case case case

1: If Xa = R, let /(α) = 0. 2: If Xa has a least element 6, let /(α) = b. 3: If the leftmost interval of Xa = (c, d), let /(α) = 4: If the leftmost interval of Xa = (—00, c), let /(α) = c — 1. 5: If the leftmost interval of X\ = (c, +00), let /(α) = c + 1. This exhausts all possibilities. Clearly / is definable and does the job.

Definable functions have a very nice application. The following theorem of Milnor ([Mi]) was first proved by geometric techniques. Theorem 2.15. (Curve selection) Let X be a definable sunset of Rn and let α be a point in the closure of X. There is e > 0 and a continuous function / : (0, e) -> Rn. Such that f ( x ) G X for all x G (0, e) and lim^o /(*) = a. Proof. Let D = {(5, x) : x G X and \x — a\ < δ}. Since R has definable Skolem functions, there is an η > 0 and a definable / : (0, η) —>• X such that/(ί) G X and \f(δ) - a\ < 0 for all δ G (0, η). By 2.13 there is an e G (0, η) such that / is continuous on (0,r/).

§3 Cell Decomposition

Let R be a real closed field. We next study the structure of semi-algebraic subsets of Rn. As a warm up we prove Thorn's Lemma. Let

(

-1 0 1

x <0 x - 0. x> 0

Theorem 3.1. (Thorn's lemma) Let / i , . . . , fs be a sequence of polynomials in R[X] closed under differentiation. For σ G {-1,0,1}S let = {x<=R:/\ βgn(/, (aO) = σ(i)}.

»=ι

15 Then each Aσ is either empty, a singleton or an open interval. Proof. We proceed by induction on s. If s = 1, then f\ must be identically zero and the theorem is true. Assume the theorem is true for s. Without loss of generality assume that /s+1 has maximal degree. Let η — σ\s. By induction, we can apply the theorem to /!,..., fs and η. Clearly Aη D Aσ. If Aη is empty or a singleton then so is Aσ. Thus we may assume Aη is an open interval / = (c, cί). Since fa = fs+\ for some i < s, and /,- does not change sign on 7, fs+ι is monotonic on /. If fs+ι does not change sign on 7, then Aσ = I or Aσ — 0. Otherwise /5+ι(α) — 0 for some a G / and Aσ = {α}, A σ = (c, α) or A σ = (α, d). The next theorem can be thought of as a higher dimensional version of Thorn's theorem. Let X = Λ Ί , . . . ,X n . Theorem 3.2. (Cylindric Decomposition) Suppose /i,... ,/5 £ R[X,Y]. There is a partition of Rn into semi-algebraic sets AI , . . . , A m such that for each i < ra, there are continuous semialgebraic functions £,^1,... ,£ijt.Ai —» 7Z such that: i) for all x G A,-, 6,1(2:) < £ί|2(a?) < < &,/.(*) and {&,ι(z),.. .,&,/»} contains the isolated zeros of the polynomials /ι(x, Y ) , . . . , /,(z, Y) [It is convenient to let ζi,o(x) = —oo and &,/,+! = +00.], and ii) if a?ι and x 2 are in A t and either there a) is a j such that and £»j(#2) = 2/2> of b) there is a j such that ίtjί^*) < Vk < i = 1,2, then 5

/\ sgn(/i(xi,yi)) = sgn(/ f (x 2 ,!fe)). i=l

[Intuitively ii) says that for x G AI sgτι(fj(x,y)) depends only on the relative position of y with respect to &,ι(x),... ,6,/,(^) ] Proof. Without loss of generality we may assume that /i,...,/, is closed under d ayLet q be the maximal degree of any /»• with respect to Y. Fix x £ Rn. If /, (a?,y) is not identically zero, it has at most g zeros. Let 2/1 < . . . < t//(a;) be the isolated zeros of/i (x,y),...,/,(x,y). Then/(x) < eg. For j = l , . . . , / ( a ? ) - 1, let Ja-j = (j/j,yj+i) and let / x>0 = (-00,yx) and 4|/(a.) = (% ,+oo). Then each f j ( x , y ) has constant sign for ?/ G /»,,-. Call this sign β j t i ( x ) . We define P^ the pattern at x to be the the s x 2/(x) + 1 matrix where the ith-row is: [βito(x),..., /?ίV(;r)(x), sgn(Λ (x, ft)),...,sgn(/,(x, yn))]. Since the entries of Px are just -1, 0 or 1, there are only finitely many (at most 35(254+1)^ possible pasterns. Moreover if P is a pattern, it is routine to show that Ap = {x : Px = P} is definable and hence semialgebraic.

16

Let AI, . . . , Am be all the nonempty Ap. For each i, let /t = l(x) for x G A t . Let &j(x) be the jth-element of {y : y is an isolated zero of some /^(x, Y)} for j = l,...,/(i). Clearly ξij is semialgebraic. It is clear from the construction that i) and ii) hold. We need only show that each ζij is continuous. Let x G Ai. Let yj = ξi,j(x). Thus some τ/; is an isolated zero of some /fc(x, y). Since the /,- are closed under ^p, we may assume that /fc(x, y) changes sign at y j . Since for sufficiently small e > 0, /^(x, y^ - c ) f k ( x , yj + e) < 0, there is a neighborhood Bj of x such that this is true for all z G Bj. Thus f k ( z , Y ) has a root in (yj — e, t/j + e) for all z £ Bj. Thus if x £ A, then for all sufficiently small £, there is an open neighborhood B of x such that if z E 5, then some /fc(z, y) has an isolated zero in (&,j (x) — £,&j(x) + 6) for j = 1, . . . , /(i). Hence if z G A, Π 5, ζitj(z) G (&j(#) — ^j&jOO + ^) Thus £t j is continuous at x. Cylindric decomposition will be our primary tool for studying semialgebraic sets. It gives an inductive procedure for building up definable sets. Definition. -A subset X of R is a 0-cell if X = {α} for some α £ R. -A subset X of R is a 1-cell if it is an open interval. - If X C Rm is an n-cell and / : X —> R is a continuous semialgebraic function, then Y = {(x,y)€lT+1.x€X,f(x) = y} is an n-cell. -If X C Rm is an n-cell, /, g : X —> R are continuous semialgebraic functions such that /(x) < #(x) for all x G X [we also allow / to be constantly +00 or g identically — oo], then y = {(x,y) G /T+1.x G *,/(*) < y < g ( x ) } is an n + 1-cell. Theorem 3.3 (Cell Decomposition) If A C Rn is semialgebraic, then A is a finite union of disjoint cells. Proof. Let X denote Xlt...,Xn. If /!,...,/, G ΛpΓ] and σG {-1,0,1}% let

Clearly for any semialgebraic set y we can find polynomials fι , . . . , fs and S C {-1,0, 1}5 such that

The theorem is proved by induction on n. By o- minimality it is true for n = l. Assume the theorem holds for n. By the above remarks it suffices to show that for /ι,...,/ 5 G R[X,Y] and σ G {-1,0, 1}5, the theorem holds for Aσ . We apply cylindric decomposition to /i , . . . , fs . This gives BI , . . . , Bm a

17 semialgebraic partition of Rn. By induction we may assume that each J3, is a cell. Let a-j = { ( * , y ) : * € f l j - , y = 6 l y(x)}

for j = 1, . . . , /(i) and let for j = 0, . . . , /(i). The Cίj and Dt',j are cells partitioning Rn+1 such that each fk has constant sign on each of the cells and Aσ is a finite union of cells of this kind. In [K-P-S] it is shown that cell decomposition holds for any o-minimal theory. We can now extend 2.13 to Rn . Corollary 3.4. If A is a semialgebraic subset of Rn and / : A —* R is semialgebraic, then there is B\, . . . , £m a partition of A into semialgebraic sets such that f\Bi is continuous for i = 1, . . . , m. Far more is true. Definition. If A is a semialgebraic subset of Rn and / : A —> R we say that / is algebraic if there is a polynomial p(Xι, ...,Xn,Y) such that p(#, /(#)) = 0 for all x £ A. Corollary 3.5. Every semialgebraic function is algebraic. Proof.

Suppose / : A —>• ίί is semialgebraic. Apply cylindric decomposition to a family of polynomials /!,...,/« £ Λpf, Y] which is closed under ^ such that the graph of / can be defined in a quantifier free way using /i , . . . , /, . Let J3ι, . . . , Bm be a partition of Rn into cells given by cylindric decomposition. On each Bi there is a j such that f\Bij — ξij and there is a p, £ {/i, . . . , /,} such that £»j(x) is an isolated zero of p(ar,Y) for all x £ £t . Let p = ΠP ' Then p ( x , / ( i ) ) f o r a l l x € A. For R we can say much more. In the above setting suppose U is an open subset of Rn contained in Bi. If x £ {/, then since ί»,j(«) is an isolated zero of

Thus the partial derivative is nonzero on all of Bi. By the implicit function theorem we see that f\U is real analytic. While "analytic functions" do not make sense in an arbitrary o-minimal structure, van den Dries [Dl] showed that in an o-minimal expansion of an ordered field then for any definable function and any n we can partition the domain so that the function is piecewise Cn

18 n

Definition. If U C R is an open semialgebraic and / : U —»• R is semialgebraic and analytic, we say that / is a Nash function. Corollary 3.5 shows that the study of semialgebraic functions reduces to the study of Nash functions. The next lemma is proved by an easy induction. For the purpose of this lemma R° = {0}. Lemma 3.6. If A is a fc-cell in R n , then there is a projection map π : Rn —*• R* such that π is a homeomorphism from A to an open set in R*. Also if k > 0, there is a homeomorphism between A and (0, l)k. By Corollary 3.6, every cell in Rn is connected. This type of result will not hold for arbitrary real closed fields R because even R need not be connected. For example, if R is the real algebraic numbers R = {x : x < π} U {x : x > π}. Let R be a real closed field. We say that a definable X C Rn is definably connectediΐtheτe are no definable open sets U and V such that UΓ\X and VΓ\X are disjoint and X C U U V. It is easy to see that in any real closed field cells are definably connected. Cell decomposition easily implies the following important theorem of Whitney. Theorem 3.7. If A C Rn is semialgebraic then A — C\ U . . . U Cm where C Ί , . . . , Cm are semialgebraic, connected and closed in A (ie. every semialgebraic set has finitely many connected components). In real closed fields we can develop a dimension theory paralleling the theory for algebraically closed fields. +

Definition. Let R be real closed and let K be a |.β| -saturated elementary extension of R. If α i , . . . , an £ K, let dim ( a i , . . . , a n /R) be the transcendence n degree of Λ ( α ι , . . . ,α n ) over R. If A is a definable subset of R defined by let κ n ^(vii iVmδ), A = {x e K : K |= φ(x,ί)}. Note that by model comκ pleteness, A does not depend on the choice of φ. We define dim (A) the κ dimension of A to be the maximum of dim (a/R) for α £ A . Our final proposition shows that this corresponds to the topological and geometric notions of dimension. Proposition 3.8. i) dim (A) is the largest k such that A contains a fc-cell. ii) dim (A) is the largest k such that there is a projection of A onto Rk with non-empty interior. iii) dim (A) = dim (V) where V is the Zariski closure of A.

19

For further information on semialgebraic sets and real algebraic geometry the reader should consult [Di] or [BCR].

§4 Definable Equivalence Relations. In algebra and geometry we often want to consider quotient structures. For this reason it is useful to study definable equivalence relations. The best we could hope for is that a definable equivalence relation has a definable set of representatives. This is possible in real closed fields. Let R be real closed. Lemma 4.1. Let A be a definanble subset of Rm+n. For α E Rm let Aa = {x £ tf1 : (α,z) E A}. There is a definable function / : Rm -> Rn such that /(α) E Aa for all α G Rm and /(α) = /(&) if Aa = Ab. We call / an invariant Skolem function. Proof. Let / be the Skolem function defined in 2.14. It is clear from the proof of 2.14 that /(α) = /(&) whenever Aa = Ab. Corollary 4.2. If E is a definable equivalence relation on a definable subset of Rn then there is a definable set of representatives. In algebraically closed fields we will not usually be able to find definable sets of representatives. For example suppose xEy <ΦΦ> x1 = y 2 , then by strong minimality E does not have a definable set of representatives. The next best thing would be if there is a definable function / such that /(x) = f ( y ) if and only if f ( x ) = f ( y ) . Our next goal is to show this is true in algebraically closed fields. Definition. Let T be any theory and let M be a suitably saturated model of T. Let X C Mn be definable with parameters. We say that 6 E Mn is a canonical base for X if and only if for any automorphism σ of M, σ fixes X setwise if and only if σ(6) = 6. We say that T eliminates imaginaries if and only if every definable subset of Mn has a canonical base. We first illustrate the connection between elimination of imaginaries and equivalence relations. Lemma 4.3. Suppose T eliminates imaginaries and at least two elements of M are definable over 0. If E is a definable equivalence relation on M n , there is a definable / : Mn -> Mm such that x E y if and only if f ( x ) - f ( y ) .

20

Proof. We first show that for any formula φ(v,ά) there is a formula ψa(v,w) and a unique 6 such that φ(v,ά) <->ψa(υ,b). By elimination of imaginaries we can find a canonical base 6 for X — {v : φ(a, v)}. Clearly X must be definable over 6. Thus there is a formula ψ(v,w) such that X = {v : ψ(v,b)}. Further there is a formula θ(w) such that 0(6) and if c φ b and 0(c), then ψ(v,c) does not define X. Let ψa(v, w) be θ(w) Λ ^(ι>, w). By compactness we can find ψiy...,ψn such that one of the ψi works for each α. By the usual coding tricks we can reduce to a single formula ψ (a sequence of parameters made up of the distinguished elements is added to the witness 6 to code into the parameters the least i such that ψi works for α). The lemma follows if we let φ(v, w) be v E w and let /(α) be the unique 6 such that v E a if and only if ψ(v, b). We will show that algebraically closed fields eliminate imaginaries. This will follow from the following two lemmas. Lemma 4.4. Let K be a saturated algebraically closed field and let X C Kn be definable. There is a finite C C Km such that if σ is an automorphism of K, then σ fixes X setwise if and only if σ fixes C setwise. Proof. Let φ(v, αi, . . . , α m ) define X . Consider the equivalence relation E on Km given by a E b & (φ(v, a) «-+ φ(ϋ, &)). Let α denote the equivalence class a/E. Any automorphism of σ fixes X setwise if and only if it fixes α. (Note: a is an example of an "imaginary" element that we would like to eliminate.) We say that an element x £ K is algebraic over α if and only if there are only finitely many conjugates of x under automorphisms which fix α. Our first claim is that there is b £ Km algebraic over α such a E b. Choose 6 such that 6 E α, and j = |{t < m : 6, is algebraic over α}| is maximal. We must show that .; = m. Suppose not. By reordering the variables we may assume that &ι, . . . , bj are algebraic over α and 6t is not algebraic over α for i > j. Let Y = {x G K :3y j + 2 .. 3yn (&ι, . . . , f y , x , y j +2, - - - ,y n ) £^ and (6ι,...,6;,«,j(, + ι , . . . , y n ) E ά}. Clearly &; +ι E Y. If Y is finite, then any element of y is algebraic over &ι, . . . ,6j,α, and hence algebraic over a. Thus by choice of 6, Y is infinite. If y is infinite, then since K is strongly minimal, Y is cofinite. In particular they is d £ K such that d is algebraic over 0. But then we can find d; +2, . . . , dm such that (bι,...,bj,d,dj+2,...,dm)/E = a and & ι , . . . , f y , d are algebraic over α. contradicting the maximality of j.

21

Let C be the set of all conjugates of b under automorphisms fixing a. So C is fixed setwise by any automorphism which fixes a. If c £ C, then c/E = a. Thus a is fixed under all automorphisms which permute C. In particular an automorphism fixes X setwise if and only if it fixes C setwise. The proof above is due to Lascar and Pillay and works for any strongly minimal set D where the algebraic closure of 0 is infinite. The second step is to show that if C C Km is finite, then there is 6 G Kl such that any automorphsim of K fixes C setwise if and only if it fixes 6 pointwise. This step holds for any field Lemma 4.5. Let F be any field. Let 61, . . . , bm G Fn. There is / and a c G Fl such that if σ is any automorphism of ί1, then σc = c if and only if σ fixes C = {&ι, . . . , ό m } setwise. Proof. This is very easy if n = 1. If 61, . . . , 6m £ F, consider the polynomial m

m—1

*=1

i=0

Then an automorphism of F fixes {δι,...,6 m } setwise if and only if it fixes (CQ, . . . , c m _ι). Here c0, . . . , c m _ι are obtained by applying the elementary symmetric functions to 61 , . . . , bm . The general case is an easy amplification of that idea. Suppose 6j = fti,...,A-

Let

for i = 1, . . . , m. Let

By unique factorization, an automorphsim of K fixes p if and only if it permutes the qi if and only if it permutes the 6, . Let c be the coefficients of p(X,Y). Corollary 4.6. (Poizat [P]) The theory of algebraically closed field eliminates imaginaries. In [M2] we give a different proof of elimination of imaginaries for algebraically closed fields using "fields of definition" from algebraic geometry (see [L2]). Suppose E is a definable equivalence relation on K. If any ~-class is infinite, then there is a unique cofinite class. Suppose all ~ classes are finite. There is a number n such that all but finitely many equivalence classes have size n. Let B

22

be the number of points not in a class of size n. A moment's thought shows that the best we can hope to do is characterize the possible values of |5|(mod n). Theorem 4.7. (van den Dries-Marker-Martin [D-M-M]) Let K be an algebraically closed field of characteristic zero and let ~ be a definable equivalence relation on K where all but finitely many classes have size n. Let B be the set of points not in a class of size n. Then \B\ = l(mod n). Albert generalized theorem 4.7. We give his argument here. Since the projective 1 1 line P is K U {00} and the Euler characteristic of P is 2, theorem 4.7 is a corollary to the following result of Albert. Theorem 4.8 Let K be a field of characteristic zero and let C be a smooth projective curve over K. If ~ is a definable equivalence relation on K where almost all classes have size n and B is the number of points not in a class of size n, then \B\ = χ(C')(mod n), where χ(C) is the Euler characteristic of C mod n. Our proof of 4.8 will use the following simple combinatorial fact. Lemma 4.9. Let ~0 and ~ι be equivalence relation on C such that all but finitely many E^-classes have size n for i = 1,2. Let Bi = {x £ C : \x/ ~< | φ n}. Suppose for all but finitely many a?, x/ ~Q= x/ ~ι, then [.Sol = |#ι|(mod n). Proof of 4.8 Suppose C C Pm. Let Co = CΓ\Km. By 4.6 there is a definable / : CQ -> K1 such that x ~ y if and only if f ( x ) = f ( y ) for x,y £ CQ. By 1.11 there is a Zariski open U C CQ and a rational p : U —* K1 such that f\U — p. Let C\ be the Zariski closure of the image of CQ under p. Then C\ is an irreducible affine curve. There is a smooth projective curve C^, an open V C CΊ, and a rational one-to-one r : V —> CΊ (see [H] for the facts about curves used in this proof). The composition τ o p maps a dense open subset of C into 62- There is a total rational g : C —>• 62 extending τ o p. There is a cofinite subset Z of C such that 0(x) — g(y) if and only if x ~ y for x, y £ Z. Consider the equivalence relation ~ι on C given by x ~ι y if and only if g ( x ) — g ( y ) . By 4.9 we may assume f

ι*^ — ^Λ .

Let (V, E, F) be a triangulation of 62 such that the set of verticies V contains Vb = {g(x) : x G B}. Let (V*,E*,F*) be the triangulation of C obtained by pulling back the triangulation of C^. Since the edges and faces do not contain images of points in B, \E*\ = n\E\ and |F*| = n\F\, while |F*| Thus

n). The situation in characteristic p is more complex.

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Theorem 4.10.([D-M-M]) Let K be an algebraically closed field of characteristic p and let ~ be a definable equivalence relation on K such that all but finitely many ^-classes have size n. Let B be the set of points not in a class of size n. i) If n < p, then \B\ = l(mod p). ii) If n = p = 2, then \B\ = 0(mod p). iii) If n = p + s where ! < « < § , then |B| φ p + l(mod n). iv) Everything else is possible. A consequence of Hurwitz theorem (see [H]) is that if X and Y are smooth projective curves and / : X —» Y is a non-trivial rational map, then the genus of Y is at most the genus of X. This has two interesting consequences for us. 1 First, in the proof of 4.9, if C = P , then the curve 62 has genus zero and we 1 may assume that €2 is P . Thus if ~ is a definable equivalence relation on K there is a rational function / : K —» K such that there is a Zariski open U C K such that x ~ y <£> f ( x ) = f(y) for all but z, y £ U (this is proved in [D-M-M] by an appeal to Luroth's theorem). Second, let C be a curve of genus g > 1. View C as a structure by taking as relations all definable subsets of Cn. This is a strongly minimal set which does eliminate imaginaries. Suppose, for example, that C C K^. Let ~ be the equivalence relation on C given by (x, t/) ~ (u, v) if and only if x = u. Then C/ ~ is essentially K. If we could eliminate imaginaries there would be a definable map /o : C/ ~—* Cn and by composing with a projection, there would be a nontrivial definable map from C/ ~ to C. As in the proof of 4.9 this induces a rational map from P1 into C, violating Hurwitz's theorem.

§5 u -stable groups. In this section we will survey some of the basic properties of u -stable groups. Comprehensive surveys of these subjects can be found in [BN], [Po3] and [NP]. Here we assume passing acquaintance with the results about u -stable theories. The reader is referred to [Bl], [Pil] and [Po4]. Definition. An ω-stable group is an u -stable structure (G, •,...) where (G, •) is a group. Lemma 5.1. (Baldwin-Saxel [BS]) An u -stable group has no infinite chain of definable subgroups. Proof. Let HQ D HI D be an infinite descending chain of definable subgroups. We can find elements {aσ : σ G 2<ω} such that i) if <τ D r then aσίϊ\σ\ C aτH\τ\, and ii) if aσιH\σ\+ι and α<7θ#|<7|+ι are distinct cosets.

24 N

This gives a countable set of parameters over which there are 2 ° types, contradicting α -stability. Definition. A group G is connected if it has no definable subgroup of finite index. Lemma 5.2. If G is an α -stable group, then there is G° a definable connected subgroup of G of finite index. Proof. If not then we can build an infinite descending sequence of finite index subgroups. We call G° the connected component of G. Note that G° is fixed by all group automorphisms of G. Definition. If A C G we say that p(v) £ Sι(A) is a generic type over A if RM(p) = RM(G). Generic types are our main tool in studying α -stable groups. We begin by summarizing basic facts about generic types. We fix G an α -stable group. Lemma 5.3. i) There are only finitely many types generic over A. ii) If 6 is generic over A and a £ A, then ab and b~l are generic over A. iii) Any element of G is the product of two generics (in an elementary extension). Proof. i) There are only finitely many types of maximal rank, ii) The maps x \-+ ax and x ι-+ x~l are definable bijections and definable bijections preserve rank. iii) Let a £ G. Let 6 be generic. Then αά"1 is also a generic and a = (ab~l)b. Lemma 5.4. An α -stable group G is connected if and only if there is a unique generic type in SΊ(G). Proof. Suppose H is a proper definable subgroup of finite index. Then each coset of H contains a type of maximal Morley rank. Thus the generic type is not unique. On the other hand suppose P i , . . . ,pn are the generic types of G. Let H = {g £ G : for all realizations b of p\ (in, say, a saturated elementary extension), gb is also a realization of pi}. We call H the left-stabilizer of p in G. claim. H is definable. There is a formula θ(v) which isolated pi from the other generic types. Then H = {g : θ(g v) £ pi}. By definability of types there is a formula dθ(w) such that G |= dθ(w) if and only if θ(g - v) £ pi. Clearly H = {g : dθ(g)}.

25

Suppose 6 realizes pi (in an elementary extension) and α E G, then ab realizes p, for some i. Thus the coset aH contains a generic. Hence H has finite index so H = G. Similarly G stabilizes each p{. A similar argument works for right stabilizers. Let a and 6 be independent realizations of p\ and p2. Let pi be the heir of Pi to G U {6} (ie. p* is the unique extension of pi to G U {&} of maximal rank). By the above arguments b stabalizes pj, thus ba realizes p* and, in particular, ba realizes pi. A similar argument (using right stabilizers) shows that ba realizes PiWe now have enough tools to prove the following theorem of Macintyre ([Mac]). Theorem 5.5. Let (#,+, •,...) be an infinite ω-stable field. Then K is algebraically closed. Proof. Suppose K is not algebraically closed. Let F be a finite Galois extension of K. There is L such that K C L C F and the Galois group of F/L is a cyclic extension of prime order q. Since L is a finite extension of K, we can interpret L in K. Thus L is ω-stable so we may, without loss of generality assume that F/K is cyclic of prime order. By Galois theory (see [LI]) F = K(&) where either q ^ p and aq £ K or q = p and ap + a £ K. We first show that (K, +,...) is connected. Suppose not. Let H be the connected component. For any α £ A", x »->• ax is an automorphism of (ίf,+) and hence preserves F. But then # is a proper ideal of K, a contradiction. Since (AT, +) is connected, there is a unique type of maximal rank. Thus there is a unique type of maximal rank in the group (K*,-,...) and hence it is connected. Consider the multiplicative homomorphism x ι-+ xn. If α is a generic of K, then, since α is algebraic over α n , RM(αn) = RM(α). Thus {xn : x £ K*} is a subgroup of K* of maximal rank. Since Kx is connected, every element of K has an nth-root in K. This rules out the case aq G K. Suppose K has characteristic p > 0. Consider the additive homomorphism x h-» xp + x. As above if α is generic, so is ap + a. Thus since the additive group is connected, for any 6 E K, there is a solution to Xp -f X = 6. This rules out the case ap + α G -K".

26

As an aside, we note the following theorem of Pillay and Steinhorn ([PS]) can be thought of as the real version of Macintyre's theorem. 1

Theorem 5.6. Let (ί ,+,-,<,...) be an o-minimal ordered field. Then F is real closed. Proof. Let f ( X ) G F[X]. Suppose α,6 G F, a < b and /(α) < 0 < /(6). Consider the set X- = {x G (α,6) : /(*) < 0} and X+ = {x G (α,6) : /(*) > 0}. Since / is continuous X~ and X+ are open. By o-minimality there is c G (α, 6) \ (X- U X+). Clearly /(c) = 0. By 2.1, F is real closed. One important problem in the model theory of groups is to understand the simple groups of finite Morley rank. Cherlin's Conjecture. Every simple group of finite Morley rank is an algebraic group over an algebraically closed field. We recall the definition of an algebraic group. Definition. An abstract variety is a topological space B with a finite open cover Uι,..., Un, affine Zariski closed sets V\,..., Vn and homeomorphisms /,- : Ui —> Vi such that if Vij = /<(t/i Π Uj) and fitj : Vij -> V$ |f is the map ft o /r 1 , then Vij is Zariski open and fcj is a morphism. If W is a second abstract variety with cover Z\,..., Zm where W^ is a homeomorphism onto an affine Zariski closed set, then h : V —> W is a morphism if all the maps hij : Vi —» Wj by gj oho /i~1 are morphisms of affine varieties. Abstract varieties are the algebraic-geometric analog of manifolds. Clearly affine and projective varieties are examples of abstract varieties, as are open subsets of projective varieties. We drop the modifier "abstract". Definition. An algebraic group is a group (G, •) where G is a variety and inverse are morphisms.

and

The standard examples of algebraic groups are matrix groups. For example consider GLn(K), the invertible n x n matrices. As the underlying set we take {(dij,b) G Kn*+l : 6det(α( f|J )) = 1}. This is a Zariski closed set in affine n 2 + lspace. The extra dimension codes the fact that the determinant is non-zero. Matrix multiplication is easily seen to be given by polynomials. Using Cramer's rule one sees that the inverse is also given by polynomials. The group law on an elliptic curve is an example of a non-affine algebraic group. It is easy to see that every algebraic group G over an algebraically closed field K is interpretable in K. Thus, by elimination of imaginaries, G is isomorphic

27

to a constructible group. A priori one might expect there to be constructive groups which are not isomorphic to algebraic groups. This is not the case. Theorem 5.7. (van den Dries [D4]) Let K be an algebraically closed field. Every constructible group over K is K-definably isomorphic to an algebraic group. Van den Dries' proof uses a theorem of Weil's on group chunks. Weil's theorem actually shows that if V is an irreducible variety and / : V x V —>• V is a generically surjective rational map such that f ( x ( f ( y , z)) = /(/(#, 2/), z) for independent generic z, y, z, then there is a birationaly equivalent algebraic group G, such that generically / agrees with the multiplication of G. Hrushovski (see [Bol] or [Po3]) gave a model theoretic proof of theorem 5.7 avoiding Weil's theorem. In [Hrl] Hrushovski proved the following result which can be though of a general model theoretic form of Weil's theorem. Theorem 5.8. (Hrushovski) Let T be an ω-stable theory. Let p £ Sn(A) be a stationary type and let / be a partial A-definable function such that i) if α and 6 are independent realizations of p, then /(α,6) realizes p and /(α,6) is independent over A from α and 6 separately, and ii) if α,δ and c are independent realizations of p, then /(α,/(&, c)) = Then there is a definable connected group (G, •) such that p is the generic type of G and if α, 6 are independent generics of G, then a - b = /(α, 6). Pillay [Pi2] proved the following o-minimal analog of theorem 5.6. Theorem 5.9. If G is a group definable in an o-minimal expansion of R, then G is definably isomorphic to a Lie Group. Finally we remark that Peterzil, Pillay and Starchenko have recently proved the following o-minimal analog of Cherlin's conjecture. Theorem 5.10. If G is a simple group definable in an o-minimal theory, then there is a definable real closed field K such that G is definably isomorphic to a group definable in K. Indeed there is an algebraic group H definable over K such that G is definably isomorphic to H°.

28

§6 Expansions and reducts of algebraically closed fields. Suppose D is a strongly minimal set. The algebraic closure relation on D has the following properties. i)XC aclpf), ii) acl(acl(X)) = acl(X), iii) if α € acl(X, 6) \ acl(X), then b G acl(Λ\ α), and iv) if a G aclpf), then there is a finite X0 C X such that α G acl(X0). We say that X C D is independent if x ^ acl(X \ {x}, for all x G X. We say that X is a 60515 for A if A C acl(X) and X is independent. A simple generalization of the arguments from linear algebra show that any two basis for A have the same cardinality. We call this cardinality dim A. Definition. We say that a strongly minimal set D is trivial if whenever A C D, then acl(A) = U acl(α). α€Λ

We say that D is modular if dim (A U J3) = dim A + dim B - dim (Λ Π B)

for any finite dimensional algebraically closed A, B C D. We say that D is locally modular if we can name one point and make it modular (this is equivalent to being make it modular by naming a small number of points). The theory of Z with the successor function x i—>• x + 1 is a trivial strongly minimal set. Here α G acl(X) if and only if α — sn(x) for some n G Z and x £ X. If V is a vector space over the rationale. The strongly minimal set (V, +) is modular. Here acl(X) is the linear span of X. We can modify this to give a locally modular example. Consider (V,/) where / is the ternary function /(#, y, z) = x + y — z. In this language, acl(X) is the smallest coset of a linear subspace that contains X. For example acl(α) = {α} and acl(α,6) is the line containing α and 6. It is easy to see that (V,/) is not modular. Let α,6, c be independent points and let d = c + b — a. Then dim (α,6, c, d) = 3 while dim (α, 6) = dim (c, d) = 2 and acl(α, 6) Π acl(c, d) = 0. On the other hand if we name 0, we are essentially back to the structure (V,+). Let K be an algebraically closed field of infinite transcendence degree. We claim that (X, +, •) is not locally modular. Let k be an algebraically closed subfield of transcendence degree n. We will show that even localizing at k the geometry is not modular. Let α, 6, x be algebraically independent over k. Let y — ax + b. Then dim (k(x, y, α, 6)) = 3 + n while dim (*(x, y)) = dim (fc(α, 6)) = 2. But acl(fc(x, y)) Π acl(£(α, 6)) = k contradicting modularity. To see this suppose d G kι = acl(k(a,b)} and y is algebraic over k(d,x). Let kι = acl(fc(cf)). Then there is p(X,Y) G ti[-X",y] an irreducible polynomial such that p(x, y) = 0.

29

By model completeness p(X,Y) is still irreducible over acl(£(α,ό))[X, Y]. Thus p(X,Y) is a(Y — aX — b) for some a £ acl(fc(α,6)) which is impossible as then a £ kι and α,6 £ kι. The geometry of strongly minimal sets has been one of the most important topics in model theory for the last decade. Much of this work was motivated by the following conjecture. Zilber's Conjecture. If D is a non-locally modular strongly minimal set, then D is bi-interpretable with an algebraically closed field. Zilber's conjecture was refuted by Hrushovski in [Hr2] (see also [BSh]). Though false Zilber's conjecture led to two interesting problems about algebraically closed fields. Expansion Problem: Can an algebraically closed field have a nontrivial strongly minimal expansion? Interpret ability Problem: Suppose D is a non-locally modular strongly minimal set interpretable in an algebraically closed field K . Does D interpret Kl In [Hr3] Hrushovski showed that there are nontrivial strongly minimal expansions of algebraically closed fields. Indeed, one can find a strongly minimal structure (F, +, ,Θ,Θ) such that (F, + ,) and (F, 0,0) are algebraically closed fields of different characteristics! Prior to Hrushovski's work several positive results were obtained. The first is an unpublished result of Macintyre. Proposition 6.1. If / : C —> C is a non-rational analytic function,then (C, +, , /) is not strongly minimal. Proof. Suppose not. By strong minimality / must have only finitely many zeros and poles. Thus (see [L3]) f ( x ) = g(x)eh^ where g is rational and h is entire. Since g is definable so is fo(x) = eh(χ\ But /0 is infinite to one, so the inverse image of some point is infinite and cofinite, contradicting strong minimality. Is this structure stable? Definition. Suppose 5 C R 2n . Let 5* = {(αi -h α 2 i , . . . , α 2 n _ι + a2ni) € Cn : ( α i , . . . , a2n) £ S}. A semialgebraic expansion of C is an expansion (C,-h, ,5*) where S C R2n is semi algebraic. There are two obvious ways to get a semialgebraic expansion. The first is to add a predicate for a set which is already constructible. The second is to add a predicate for R. The next theorem shows that these are the only two

30

possibilities. Since the reals are unstable, this shows in a very strong way that there are no nontrivial strongly minimal semialgebraic expansions of C. Theorem 6.2([M1]) If A — (C, +, , 5*) is a semialgebraic expansion then R is A- definable. We will prove this theorem using four lemmas. The first lemma is the basic step. We omit the proof, but remark that it works in a more general setting. Lemma 6.3 If A = (C,+, ,S"*) where S* is an infinite coinfinite subset of C definable and S is definable in an o-minimal expansion of R, then R is definable in A. We give here a proof of 6.2 from 6.3 which is more direct that the original argument from [Ml] and is based on an argument from [Hr3]. Assume that 5* is not constructive and R is not definable. By 6.3 we may assume that every ^-definable subset of C is finite or cofinite. Since C is uncountable, this suffices to show that the structure A is strongly minimal. The next lemma of Hrushovski shows that we may assume that S* C C 2 . It replaces a less general inductive argument using Bertini's theorem. Lemma 6.4. If X C Cn is non-constructible and A — (C,-f, ,S"*) is strongly minimal, then there is a non-constructible Λ-definable h : C —» C. Proof. Without loss of generality assume that every definable subset of Cm is constructive for m < n. Let Xa = {x G C""1 : (α,z) G X} for a G C. Each Xa is constructive. Thus for each α G C we can find a number m α , a formula Φa(v\ι - > ^n-i) MI, . . . , wm<J in £Γ, and parameters ba G C m * such that x G Xa &Φa(x,ba). Since A is saturated, compactness insures there are formulas φι>...yφk such that for each α at least one of the φi works. By standard coding tricks one formula φ(v\,..., vn, wι,..., wm) suffices. Define an equivalence relation E on C m by

a E b & MX (φ(x, a) <-> φ(x, 6)). By elimination of imaginaries, there is a constructive function g : Cm —* C1 such that a E b if and only if g(ά) = g(b). Define / : C -* C' by /(α) = y <-> V6 (Vz (φ(z, 6) <-> z G Xa) -> g(b) = y).

31 Clearly / is definable and (α,y) G X & 36 (g(b) = /(α) Λ φ(y,b)). Since 0 is constructible and X is not, / is not constructible. Let ft be a non-constructible coordinate of /. Lemma 6.5. Suppose S* is semi algebraic and non-constructible. Let ft be as in 6.4 and let H be it graph. There is an irreducible curve C such that H Π C and C\H are infinite. Proof. In our setting ft is semi algebraic. Consider the following two predicates over R:

R 0 (x, y) <-> 3z h(x) = y + zi RI(X, z) <-+ 3y ft(x) = y + zi 2

Let Vi be the Zariski closure of Rt in R . Each Λ, is one dimensional, thus, by 3.8, each Vi has dimension one. In particular, since each one dimensional irreducible component of Vi is a curve, we can find non-trivial polynomials f i ( X , V) such that

Λt (χ,y)-»Λ (χ,y) = 0.

We now move back to C. Let 4

AO = {(*, y, *, w) G C : /0(x, y) = Λ(x, z) = 0 Λ ty = y + *t}.

Let Λ = {(a:, w) : 3y, 0 (x, y, z, ιy) G -Ao} Clearly ^4 and ^4o are constructible and one dimensional. Moreover (x, Λ(x)) G A for x G R. Thus by strong minimality (x,Λ(x)) G A for all but finitely many x G C. Thus there is C an irreducible component of the Zariski closure of A such that(x,Λ(x)) G C for all but finitely many x G C. Since ft is not constructible, for a generic x there is more than one y such that (x, y) G C. Thus H Γ\ C and C\ H are infinite. The following lemma of Hrushovski finishes the proof. In [Ml] this was proved in the semialgebraic case by appealing to a weak version of the RiemannRoch theorem. Lemma 6.6. Let Λ = (C,+, -,X) be a nontrivial expansion of C, where X is an infinite coinfinite subset of an irreducible curve C. Then A is not strongly minimal. Proof. We assume A is strongly minimal. The proof breaks into cases depending on the genus of C. If C has genus 0 there is a Zariski open U C C and a one to one rational p : U —> C. Clearly p(X) is an infinite coinfinite subset of C. Any curve is birationally equivalent to a smooth projective curve. Since projective curves can be interpreted in C and rational maps are definable, we may, without loss of generality, assume that C is a smooth projective curve.

32

If C has genus 1 , then there is a morphism 0 : C x C —> C making C a divisible abelian group (see [H] or [F]). We consider the ω-stable group Q = (C, φ, X). The sets X and C \ X are Morley rank one subsets of C. Thus there are distinct types of maximal Morley rank. Hence, by 5.4, Q has a definable subgroup of finite index. But a divisible abelian group has no finite index subgroups, a contradiction. If C has genus g > 1, we must pass to J(C) the Jacobian Variety of C. We summarize the facts we use (see [L2] or [Mu]). (Note: If C has genus 1, then i) J(C) is an irreducible g dimensional variety. ii) There is a rational p : C9 —» J(C) which takes g independent generic points of C to the generic of J(C). iii) There is a morphism φ : J(C) x J(C) —> J(C) making J(C) a divisible abelian group. By ii) p(X*) and p((C \ X)9) both have Morley rank g. Thus, as in the genus 1 case, we are lead to a contradiction That concludes to proof of theorem 6.2. Here are three natural open questions related to the expansion problem. Let K be algebraically closed. 1) Is there a non- trivial infinite multiplicative subgroup G of K such that (K, +, , G) has finite Morley rank? 2) Suppose K has characteristic p > 0. Is there a non-trivial infinite additive subgroup G of K such that (K, + , ,G) has finite Morley rank? The answer is no if K has characteristic zero ([Po3]). 3) Suppose K has characteristic p > 0. Is there an undefinable automorphism σ of K such that (K, +, , σ) is strongly minimal? The Interpretability Problem is still open. An important special case was proved by Rabin ovich [R]. Theorem 6.7. Let K be algebraically closed and let X ι , . . . , X n be constructible. If Ω = (K,Xι, . . . ,Xn) is non-locally modular, then Ω interprets and algebraically closed field isomorphic to K. Prior to Rabin ovich 's theorem results were know in some special cases. Theorem 6.8. (Martin[Maj) Let p : C —» C be a non-linear rational function. Then multiplication is definable in (C,+,p. The next result gives a complete description of reducts of C that contain +. n For each α G C, let λ α (x) — ax. We say that a subset X C C is module definable if it is definable in the structure (C,+,λ α : α G C). If X is module definable, then there is no field definable in (C, +,JΓ). This is the only restriction.

33

Theorem 6.9. (Marker-Pillay [MP]) If X is constructible but not module definable, then multiplication is definable in (C,+,X). There are three steps to the proof. The main step is due to Rabinovich and Zilber. The proof below, follows their basic ideas, but is simplification of their original argument. Theorem 6.10. If C is an irreducible non-linear curve, then there is a field interpret able in A = (C, +, C). Proof, (sketch) Without loss of generality we assume that (0,0) £ C. If p £ C, let Cp be the curve obtained by translating p to the origin. If p is a nonsingular point on C, let m(p) be the slope of the curve at p. Let C and D be curves through the origin. We define two new curves

and

C®D = {(*,*) : By ( x , y ) E C Λ (y, z) E D}. If C and D have slopes m and n at the origin, then, if they are smooth at (0,0), C Θ D and C 0 D respectively have slopes ra + n and mn at the origin. We show how to define a "fuzzy" field structure on C. Let α and 6 be independent generic points of C. There is a point d on C such that m(α)+m(6) = m(d). We show that d is algebraic over a and 6. Let D be the curve Ca Θ CV There is a number s such that |Car Π D\ — s for all but finitely many points x £ C. We claim that \Cd Π D| < s. Clearly Cd and £) have the same slope at (0,0). Thus the origin is a multiple point of intersection. If we make a small translation along the curve, the point of intersection at the origin will become two or more simple points of intersection. Moreover, no new multiple points of intersection will form. Thus the number of points of intersection goes up. Since this translation was generic, we must have originally had fewer that the generic number of point of intersection. Similarly if m(α)m(6) — m(e), then e is algebraic over a and 6. Thus there are formulas A(x, y, z) and M(x, y, z) such that if α and 6 are independent generic points on C, then {z : A(a,b,z}} and {z : M(α,6,z)} are finite, if m(α) + m(6) = ra(d), then A(atb,d) and if m(ά)m(b} = m(e) then M(α,6, e). This is what we call a "fuzzy field". Using Hrushovski's group configuration (see [Bo2]) one sees that in an ω-stable fuzzy field one can interpret a field. The proof of 6.9 also works if C is a strongly minimal set (in (C,+,C)) which is a finite union of non-linear curves. The next lemma shows that this is the only case we need consider.

34

Lemma 6.11. If X is a constructible set which is not module definable, then 2 there is a strongly minimal subset of C which is a finite union of non-linear curves. The proof is an inductive argument using Bertini's theorem. Theorem 6.10 now follows from the next lemma. Lemma 6.12. If K is an algebraically closed field of characteristic zero and A = (K, +,...) is a reduct in which there is an infinite iterpretable field F, then • is definable in A. Proof. Since A is strongly minimal, K is contained in the algebraic closure of F. By a theorem of Hrushovski (see for example [Pi3] ) or [Po3]) there is a + + proper definable normal subgroup N of K such that K /N is definably (in n A) isomorphic to a group G contained in F . Since K as characteristic 0, by a result of Poizat (see [Po3]) N = {0}, so G = K+. It is known ([Po3]) that any infinite field F interpretable in a pure algebraically closed field K is definably (in K) isomorphic to K. It then follows that F is also a pure algebraically closed field. In out case this implies that the group G is definable in F. Since G is definable in F, by theorem 5.6, G is definably isomorphic to an algebraic group over F. It is easy to see that G is one dimensional and connected. It is well known (see [Sp]) that any such group is either an elliptic curve or isomorphic to the additive or multiplicative group of the field. Since G is torsion free it must be isomorphic to the additve group of the F. In particular in A, there is a definable isomorphism between K+ and F+. We identify F+ and K* and define ® a multiplication on fί, induced by the multiplication on F. Let B = {a £ K : Vz,y (x ® (ay) = a(x ® y)}. We claim that B = K. Clearly all the natural numbers are in B. Thus B must be cofinite. Since any element of K can be written as the sum of two elements of B, it is easy to see that B — K. Let σ be the map x ι-» 1 0 x. It is easy to see that σ is definable in A and xy = z & σ~1(x) 0 σ~l(y) - σ~l(z). So multiplication is definable in A. One could also ask about analogous problems for R. Some of the most important recent work in model theory has been the study of o-minimal expansions of R. The most exciting breakthrough was Wilkie's proof that the theory of (R,+, -,ex) is model complete and o-minimal. We refer the reader to [W], [MMD] and [DD] for more information on this subject. The problem of additive reducts was solved in the series of papers [PSS], [Pe] and [MPP].

35

Theorem 6.13 i) (Pillay-Scowcroft-Steinhorn) If B C Rn is bounded then multiplication is not definable in the structure (R, +, <,B,\ a : α G R). ii) (Peterzil) If X C Rn is semialgebraic but not definable in (R,+,< , |[0,1]2, λ α : a G R),then multiplication is definable in (R, +, <,X). iii) (Marker-Peterzil-Pillay) If X C Rn is semialgebraic but not definable in (R, +, <, λ α : α G R), then |[0,1] is definable in (R, +, <,X). Recently Peterzil and Starchenko ([PeS]) have proved the o-mimmal analog of Zilber's conjectue. Let M be an o-minimal structure. We say that M is nontrivial at α if there is an open interval / containing α and a definable continuous F : I x I —> M such that for all b G / x I the functions x *-> F(b, x) and x π-» F(z, 6) are strictly monotone. Otherwise we say that M is trivial at α. Theroem 6.14. (Perterzil-Starchenko) Let M be an Ni-saturated o-minimal structure. If α G M, then exactly one of the following hold: i) M is trivial at α, ii) the sturucture that M induces on a neighborhood of α is a reduct of an ordered vector space, or iii) the structure that M induces on a neighborhood of α is an o-minimal expansion of a real closed field.

References

[A] J. Ax, The elementary theory of finite fields, Ann. Math. 88 (1968). [B-l] J. Baldwin, Fundamentals of Stability Theory, Springer Verlag (1988). [B-2] J. Baldwin and N Shi, Stable generic structure, Ann. Pure and Applied Logic (to appear). [B-S] J. Baldwin and J. Saxl, Logical stability in group theory, J. Australian Math. Soc., 21, (1976). [B-C-R] J. Bochnak, M. Coste, and M. F. Roy, Geometric Algebrique Reelle, Springer Verlag (1986). [B] A. Borel, Injective endomorphisms of algebraic varieties, Arch. Math. 20 (1969). [BN] A. Borovik and A. Nesin, Groups of Finite Morley Rank, Oxford University Press (1994). [Bo-1] E. Bouscaren, Model theoretic versions of Weil's theorem for pre-groups, in [N-P].

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[Bo-2] E. Bouscaren, The group Configuration-after E. Hrushovski, in [N-P]. [C-K] C.C. Chang and H.J. Keisler, Model Theory, North Holland (1977). [(.)] G. Cheήm.Model Theoretic Algebra, Springer Verlag, SLN 521, (1976). [Di] M. Dickman, Applications of model theory to real algebraic geometry, in Methods in Mathematical Logic, Springer Verlag, SLN 1130, (1985). [D-D] J. Denef and L. van den Dries, p-adic and real subanalytic sets, Ann. Math. 128 (1988). [D-l] L. van den Dries, Remarks on Tarski's problem concerning (R, 4-, ,exp), in Proc. Peano Conference 1982, North Holland (1984). [D-2] L. van den Dries, Alfred Tarski's elimination theory for real closed fields, JSL 53, (1988). [D-3] L. van den Dries, Algebraic theories with definable Skolem functions, JSL 49 (1984). [D-4] L. van den Dries, WeiPs group chunk theorem: a toplogical setting Illinois J. of Math. 34(1990). [D-Mac-M] L. van den Dries, A. Macintyre, and D.Marker, The elementary theory of restricted analytic fields with exponentiation, Annals of Math. 140 (1994), 183-205. [D-M-M] L. van den Dries, D. Marker and G. Martin, Definable equivalence relations on algebraically closed fields, JSL 54 (1989). [F] W. Fulton, Algebraic Curves, Benjamin Cummings, (1974). [H] R. Hartshorne, Algebraic Geometry, Springer Verlag (1977). [Hr-1] E. Hrushovski, Contributions to Stable Model Theory, Ph.D. dissertation, University of California, Berkeley (1986). [Hr-2] E. HrushovskijA new strongly minimal set, Ann. Pure and Applied Logic 62 (1993). [Hr-3] E. Hrushovski, Strongly minimal expansions of algebraically closed fields, Israel J. Math 79 (1992). [K-P-S] J. Knight, A. Pillay and C. Steinhorn, Definable sets in ordered structures II, Trans. Amer. Math. Soc. 295 (1986). [L-l] S. Lang, Algebra, Addison-Wesley (1971). [L-2] S. Lang, Introduction to Algebraic Geometry, Interscience, (1958). [L-3] S. Lang, Complex Analysis, Addison-Wesley (1977). [Mac] A. Macintyre, On u>ι -categorical theories of fields, Fund. Math. 71 (1971). [M-M-D] A. Macintyre, K. McKenna and L. van den Dries, Quantifier elimination in algebraic structures, Adv. in Math. 47 (1983). [Ml] D. Marker, Semi-algebraic expansions of C, Trans. Amer. Math. Soc. 320 (1990).

37

[M2] D. Marker, Model theory of differential fields, this volume. [M-P-P] D.Marker, Y. Peterzil and A. Pillay, Additive reducts of real closed fields, JSL 57 (1992). [M-P] D. Marker and A. Pillay, Reducts of (C,+,.) which contain +, JSL 55 (1990). [Ma] G. Martin, Definability in reducxts of algebraically closed fields, JSL 53 (1988). [Mi] J. Milnor, Singular points of Complex Hypersurfaces, Princeton University Press (1968). [Mu] D. Mumford, Abelian Varieties, Oxford University Press (1970). [N-P] A. Nesin and A. Pillay, The Model Theory of Groups, Notre Dame Press (1989). [Pe] Y. Peterzil, Ph.D. thesis, University of California Berkeley, 1991. [Pe-S] Y. Peterzil and S. Starchenko, A trichotemy theorem for o-minimal structures (in preparation). [Pi-1] A. Pillay, Introduction to Stability Theory, Clarendon Press (1983). [Pi-2] A. Pillay, Groups and fields definable in O-minimal structures, J. Pure and Applied Algebra 53 (1988). [Pi-3] A. Pillay, On the existence of [delta]-definable normal subgroups of a stable group in [N-P]. [Pi-4] A. Pillay, Model theory of algebraically closed fields, Stability theory and algebraic geometry, an introduction (proceedings of Manchester workshop) E. Bouscaren and D. Lascar ed., preprint. [P-S] A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Trans. Amer. Math. Soc. 295 (1986). [P-S-S] A. Pillay, C. Steinhorn and P. Scowcroft, Between groups and rings, Rocky Mountain J. 19 (1989). [Po-1] B. Poizat, Une theorie de Galois imaginaire, JSL 48 (1983). [Po-2] B. Poizat, An introduction to algebraically closed fields and varieties, hi [N-P]. [Po-3] B. Poizat, Groupes Stables, Nur Al-Mantiq wal-Ma'rifah, (1987). [R] E. Rabinovich, Interpreting a field in a sufficiently QMW Press (1993).

rich incidence system,

[Sp] T.A. Springer, Linear Algebraic Groups, Birkhauser (1980). [W] A. J. Wilkie, Some model completeness results for expansions of the ordered field of reals by Pfaffian functions and exponentiation, Journal AMS (to appear). [Z-R] B. Zilber and E.F. Rabinovich, Additive reducts of algebraically closed fields, (1988) manuscript (in Russian).

Model Theory of Differential Fields David Marker University of Illinois at Chicago §1 Differential Algebra. Throughout these notes ring will mean commutative ring with identity. A derivation on a ring R is an additive homomorphism D : R —> R such that D(xy) = xD(y) + yD(x). A differential ring is a ring equipped with a derivation. Derivations satisfy all of the usual rules for derivatives. Let D be a derivation on R. Lemma 1.1. For all x G R, D(xn) = nxn~lD(x). Proof. By induction on n. ^(x1) = D(x). D(xn+1) = D(xxn) = xD(xn) + xnD(x) = nxnD(x) + xnD(x) = (n + l)xnD(x). Lemma 1.2. If 6 is a unit of R, D(f ) = Proof. U(α) Thus examples. 1) (trivial derivation) D : R -> {0}. 2) Let C°° be the ring of infinitely differentiate real functions on (0, 1) and let D be the usual derivative. 3) Let U be a nonempty connected open subset of C. Let Ou be the ring of analytic functions / : U —> C and let D : Ou -+ Ou be the usual derivative. [Note: Ou is an integral domain, while the ring of C°° functions is not.] Similarly the field of meromorphic functions on U is a differential field. In appendix A, we show that every countable differential field can be embedded into a field of germs of meromorphic functions. 1 1 4) Let α G R. Let D : R[X] -> R[X] by £>(£ αt-X ') = α(£ iαjX '- ). [Note: If α = 1, then D is ^.]

39

5) Let DQ : R -» R be a derivation. We form R{X}, the ring of differential polynomials as follows. R{X} = R[XQ,Xι,. . .]. Let D extend DQ by D(Xn) = l

We identify X0 with X and Xn with χ(n), the n th derivative of X.

Definition. If D is a derivation on Λ, we let CR denote the kernel of D. We call CR the constants of R. (If no ambiguity arises we will often drop the subscript R). -C is a subring of R. Moreover if 6 £ C is a unit in R and α £ C then | is in C. In particular, if R is a field then so is C. -If a G C, then D(αa?) = α/)(ar), thus D is C-linear. Our first goal is to develop the basic ideal theory for differential ideals. We will be studying K C L where K and L are differential fields. If α G £ we will want to consider the ideal of differential polynomials over K which vanish at α. Definition. We say that an ideal I C R{X} is a differential € /.

ideal if for all / £ 7,

In general if If C L and α 6 L then the ideal {/(X) G # {X} : /(α) = 0} is a prime differential ideal. For f ( X ) G R{X}, we let (f(X)) be the differential ideal generated by f ( X ) . Even if f ( X ) is irreducible, (f(X)) may not be prime. For example let f ( X ) = (X11)2 - 2X' . Then D(f) = 1X"(X"' - 1) is in but neither 2X" nor X'" - 1 is in Definition. If f ( X ) G Λ{X} \ R, the order of / is the largest n such that X ( n ) occurs in /. (For completeness if f / G R we say / has order -1.) If / has order n we can write »=0 n

where 0, G R[X,X*',... ,X( ~1)]. If gm φ 0, we say that / has degree m. We say that f ( X ) is sπnp/erthat g ( X ) and write / 0. The separant of / is Of

For example if f ( X ) = (X")2 - 2X' , then β(X) = IX". If f ( X ) = Σ,Γ=o9i(X, .,X
i=0

So s(X) < f ( X ) .

40 k

Definition. For f ( X ) G R{X} let /(/) = {g G R{X} : s g G {/> for some k}. We will show that if R is a differential field and / G Λ{^} is irreducible, then /(/) is a prime differential ideal and that every prime differential ideal is of this form. Lemma 1.3. /(/) is a differential ideal. Proof. Clearly R { X } I ( f ) C /(/). If sng0,smgι G (/), and n < m, then sm(g0 + 9ι) € {/). Thus /(/) is an ideal. If sng G {/), then D(sn+lg) G (/). But D(^+1(7) = (n+l}sngD(s)+sn+lgf. Hence sn+lg' G (/). Thus if g G /(/), then g1 G /(/). The following division lemma is central to our analysis of differential ideals. For the rest of this section we will consider the case that is most important to us. We assume that R is a differential field K of characteristic zero. (The next lemma is false if K has characteristic p > 0.) Lemma 1.4. If / is irreducible of order n and g G {/) \ {0}, then g has order at least n and if g has order n, then / divides g. Proof. Let s be the separant of /. We need the following claim. claim: We can write /« = βX(n+0 + /,(X, . . . , X ("+'-1)), for / > 1. Let / = Σ™ o hi (X(n))', where Λ, has order at most n - 1. Then

«=o

(n) 1

where /i = £ Λ (X ) '. Thus the claim is true for / = 1. n+l Given /W = sX( ^ + //, where // has order at most n + / — 1, / > 1, we 1 n n / 1 have /C+ ) = s'X( +0 + 5X( + + ) + //. Let //+1 = // + 5'χ("+0. Then //+1 /+1 n l has order at most n + / and /( ) = Sχ( + +^ + //+1. Let 0 = αo/ + . . . + <*kf^ If fc = 0, the lemma holds, so we assume fc > 1. Assume
We next replace all instances of x(»+*-i) by Continuing we find an m and c G K{X} such that smg = cf. The degree of 5 is less than the degree of /. Thus / does not divide s. Since / is irreducible, / divides g. In particular, g has order exactly n.

41 m

Repeating the previous proof starting with s g, we can prove the following lemma. Lemma 1.5. Let / be irreducible of order n and let g G /(/) \ {0}. Then g has order greater than or equal to n and if g has order n, then / divides g. Lemma 1.6. Let / be irreducible of order n. For any differential polynomial g, m we can find g\ of order at most n such that for some m, s g — g\ (mod (/)). Proof. Suppose g has order n + fc, where k > 1. Suppose the lemma is true for all h o^i has order at most n, lemma 1.5 implies f\VQV\. Since / is irreducible f\vQ or /|vι. If/|v», then s m 'u, G {/) and u, G /(/)• Lemma 1.8. Every nonzero prime differential ideal is of the form /(/) for some irreducible /. Proof. Let / be a prime differential ideal. Let / G / be irreducible such that there is no g G / with g φ 0 and g •< /. We call / a minimal polynomial of /. We claim that / = /(/). Suppose g G /(/) and smg G (/) C /. Since / is prime and s £ /, g G /. Thus /(/) C /. Let g G /. Let g\ have order at most the order of / and m be such that smg = 0ι(mod {/)). Let d be the degree of /. Using the division algorithm we can write gι = af + ri, where α, ri G K(X ... χ(n~l))[χ(n)] and TI has degree < d. Clearing denominators, there are αι,α2,Γ2 G K[X,.. .X^] such that r2 has degree < d, a\ is of order < n and a\g\ = α2/ + Γ2 Since ^ G / and (/) C /, flΊ G /. Thus r2 G /. But r2
42

Otherwise a is differentially transcendental. [Warning: differentially algebraic does not imply algebraic in the model theoretic sense, as differential equations usually have infinitely many solutions.] We let K(a) denote the differential field generated by a over K. Lemma 1.9. If L/K are differential fields and α G L then RD(I(a/K)) is equal to the transcendence degree of K(a)/K. Proof. If I(a/K) = {0}, then K(a) is isomorphic to K(X^^Xι,X ^, . . .), a purely transcendental extension of K. Thus K(a)/K has transcendence degree ω. If not we can assume I(a/K) = /(/) where / is a minimal polynomial. Then / has order RD(I) — n. Clearly or, α/, . . . , α^""1) are algebraically independent over K, so the transcendence degree is at least n. It is also clear that a^ depends on α, α', . 5 α^"1) over K. For all k > I we can write /(*) = sXn+k + /&, where fk has order < n + k (this is the claim in the proof of lemma 1.4). Then /<*) 6 /(/), thus / (fc) (α) = 0 for all k > 1. So /<*>(*) = s(α)α(n+*) + Λ(α, . . . , α^*'1)). Thus α(n+*) depends on α, . . . ,a(n+k~1^ over K. Thus, by induction, α,. ..,e*( n ~ 1 ) is a transcendence base for K(a)/K. So K(a)/K has transcendence degree n. Note that we have shown that in the later case

We next show differential prime ideals extend when we extend the base field. Lemma 1.10. Suppose L/K are differential fields. Let / £ K{X} be irreducible and let /i G L{X} be an irreducible factor of / in L{X}. Then I κ ( f ) IL(fι)nK{X}. Proof. Suppose / has order n. If f factors in L{X}, then / factors in L[X, X1 , . . . ,χ(n^]. Moreover any irreducible factor must have order n, since whenever k C / are fields of characteristic zero, / G k[X] is irreducible and Xn occurs in /, then Xn occurs in any irreducible factor of / in l[X]. (This is an interesting exercise in Galois theory). Let Sf and s/x be the separants of / and f\. Suppose g G IL(/I) Π K{X}> Let 0ι be of order at most n such that for some m s™g = g\ (mod {/)). Then sjg = 0ι (mod (/i)) and g1 G /L(/I) Thus fι\gι. Since g1 G K{X}, all conjugates of f\ (over the algebraic closure of K) divide g\. Thus f\g\. So 9 € /*(/). Suppose gf 6 I κ ( f ) Say s^
43 If /2 G /L(/I), then, by 1.4, /ι|/2, a contradiction.

Our next goal is to prove a version of Hubert's Basis Theorem for differential ideals. Strictly speaking this is false. Even in K{X} we do not have ACC for differential ideals. For example consider the ideals, IQ C h C . . ., where:

For the rings we care about we will be able to prove ACC for radical differn ential ideals. Recall that if / is an ideal, then \/7 = {α : 3n a G /}. We say that / is a radical ideal if / = ^/ϊ. Let R be a differential ring. Lemma 1.11. If I is a radical differential ideal and ab G 7, then aD(b) G / and D(a)b G /. Proof. If ab G /, then aD(b) + bD(a) G /. Multiplying by D(a)b we see that D(a)D(b)ab + (D(a)b)2 G /. Since / is radical, D(a)b G I. Similarly aD(b) G /. Lemma 1.12. Let / be a radical differential ideal, let S C R be closed under multiplication and let T = {x G R : xS C /}. Then T is a radical differential ideal. Proof. Clearly T is an ideal. If xS C /, then, by lemma 1.11, D(x)S C I. Thus T is a differential ideal. Suppose xn G T. Then for all s G 5, xns G /. In particular for all s G 5, xnsn G /. Since / is radical, for all s G 5, zs G /. Thus x G T. For any 5 C Λ, let {5} denote the smallest radical differential ideal containing 5. Lemma 1.13. a{S] C {aS}. Proof. By lemma 1.12, T = {x : ax G {aS}} is a radical differential ideal. Since S C T, {5} C T. Lemma 1.14. Let S,T C R. Then {S}{T} C {5T}. Proof. By the previous lemma {x : x{T} C {ST}} contains {S}. Lemma 1.15. Let R D Q be a differential ring. If 7 is a differential ideal, then \/7 is a radical differential ideal. Proof. Suppose αn G /. We will prove by induction that an~k D(a)2k~l G /.

44 n

n

n

n

l

1

We know D(α ) G I. But D(a ) = na ~ D(a). Since Q C Λ, α - D(α) G 7, so the claim is true for k = 1. n k 2k 1 Suppose a - D(a) ' G 7. Then n

k

2k

(n - k)a -( +^D(a)

n

k

2k

2

+ (2k - l)a ~ D(a) ~ D(D(a)} G /.

Multiplying by by D(ά), we see that n

(n - i)α -< n

k

fc+1

2

>D(α) *

2k

l

+1

n

k

2k

l

+ (2k - l)a ~ D(a) ~ D(D(a)) G 7. n

+1

2

But (2k- l)a - D(a) - D(D(a)) G 7. So (n- t)α -<* )D(α) * n +1 +1 Λ 2 Q , α -(* )J5(α)* G 7. 2 1 Thus T^α) "- G 7, so £>(α) G V7.

+1

G /. Since

We can now prove the relevant version of Hubert's Basis Theorem. We say that a radical differential ideal I is finitely generated if there are β\ . . . βn G 7 such that / = {/?ι,. . . , βn} It is easy to see that R has ACC on radical differential ideals if and only if every radical differential ideal is finitely generated. Theorem 1.16 [Ritt-Raudenbush Basis Theorem]. Let R D Q be a differential ring such that every radical differential ideal is finitely generated. Then every radical differential ideal in R{X] is finitely generated. Proof. Suppose not. By Zorn's lemma there is a non-finitely generated radical differential ideal 7 which is maximal among the non-finitely generated radical differential ideals. We claim that 7 is prime. Suppose ab G 7, a 0 /, and b £ I. Then {α,7} and {6,1} are larger radical differential ideals and hence finitely generated. Let ci, . . . , c r , di, . . . , ds G I be such that {α, /} = {α, ci, . . . , cr} and {6,7} = {6,
In this case {α,,?} = {a,6i,j}.] Thus {α,7}{7,6} C jα6, . . . ,c r c/ 5 } C 7, by lemma 1.14. If z G 7, then z2 G {α,7}{6,7} which is contained in {αδ, . . . ,c r d 5 }, a radical ideal. Thus z G {α&, . . . , crds}, so {α6, . . . , crds} = I. Since 7 is not finitely generated we have a contradiction. Thus 7 is prime. To complete the proof we need the following stronger form of lemma 1.6: Lemma 1.17. Let 7? 3 Q be a differential ring and let / G R{X} \ R be irreducible. Suppose f ( X ) = Σ?=o ai(X^)*ι where each α; has order at most n — 1. Let s be the separant of /. For any g G R{X} there is
45

standard division algorithm for polynomials we can find g\ of degree < d such l that a dg2 = 9ι (mod {/)). We return to the proof of 1.16. We have I a non-finitely generated differential prime ideal. By assumption 7 Π R is finitely generated. Let J be the finitely generated radical differential ideal of R{X} generated by 7 Π R. Let f ( X ) £ I — J be of minimal order and d degree. Say f ( X ) = a(χW) + / 0 (X), where / 0 (X) < f ( X ) . If α € /, then /o £ 7, contradicting the choice of /. Thus a £ I. Further, s, the separant of /, is not in 7. If s £ 7, then, since s
C {αs7,7cι,...,7c m } C {J,/,cι,...,c m } C 7. If z £ 7, then z £ 72. Thus z1 £ {J,/,cι . . . ,cm}. Since this is a radical ideal, z £ { J, /, GI . . . cm}. Thus 7 is finitely generated. 2

Let fc be a differential field. We say that X C kn is D-closed if there are /i j j /m € k{X] such that

The basis theorem insures that the intersection of any collection of D-closed sets is equal to the intersection of a finite subcollection. Thus the 7)-topology is Noetherian. The next theorem gives the differential version of primary decomposition in Noetherian rings. Theorem 1.19 [Decomposition Theorem]. Let R be a differential ring with ACC on radical differential ideals. Any radical differential ideal is the intersection of a finite number of prime differential ideals. Proof. Suppose not. By ACC there is a radical differential ideal 7 which is not the intersection of finitely many prime differential ideals and is maximal with this property. As 7 is not prime, we have ab £ 7, α,6 ^ 7. Then {7,α} and {7,6} are intersections of finitely many prime differential ideals. Note that {7,α}{7,6} C {α&,7} C 7. For c £ {7,α}Π{7,6}, c2 £ 7, so c £ 7. Thus {7, α} Π {7, 6} = 7, and 7 is a finite intersection of prime differential ideals.

46

As usual there is a unique irredundant representation of I as a finite intersection of prime differential ideals. We say that a D-closed set is irreducible if it can not be written as the union of two proper D-closed subsets. The decomposition theorem implies that any D-closed set is a finite union of irreducible D-closed sets.

References For the most part the work in this section is due to Ritt. Kaplansky's monograph Differential Algebra is an excellent reference for the material in this section. It is very thin and elegantly written. In particular our treatment of the Ritt basis theorem is taken from there. Kolchin's Differential Algebra and Algebraic Groups is encyclopedic but notationally dense. Buium's Differential Algebra and Diophantine Geometry and Magid's Lectures on Differential Galois Theory are two excellent recent references. Much of the basic material on differential polynomial rings can also be found in Poizat's book Cours de Theorie des Modeles.

§2 Basic Model Theory of Differentially Closed Fields. We begin by defining the theory of differentially closed fields (DCF). Let £ be the language with binary function symbols + ,-,—, unary function symbol D, and constant symbols 0 and 1. DCF is axiomatized as follows: i) axioms for algebraically closed fields of characteristic zero ii) Vx, y D(x + y) = D(x) + D(y) iii) Vx, y D(xy) = xD(y) + yD(x). iv) For any non-constant differential polynomials f ( X ) and g(X) where the order of g is less than the order of/, there is an x such that f ( x ) = θΛ 0. This theory is much less well behaved (see [Wood]). Henceforth all fields will be assumed to have characteristic 0. Suppose K is a differentially closed field. Then as a pure field (!£,+,•) is algebraically closed. Moreover the next lemma shows that the field of constants is also algebraically closed. To avoid confusion between the field theoretic and model theoretic notions of "algebraic", we say that α is strongly algebraic over k if there is a polynomial p(X) £ k[X] — {0} such that p(a) = 0. [In §5 we will give the precise relation between algebraic and strongly algebraic.]

47

Lemma 2.1. Let K be a differentially closed field. If α G K is strongly algebraic over C the field of constants, then α G C.

Proof. 1

Let ppQ — Σ£Lo δ*-^ ke the πώώna l polynomial of α over C. Since p(α) - 0, D(p(a)) = 0. But D(p(α)) = (Σ!£?(i + l)ft,-+ia'')I>(<0- Since p is the minimal polynomial of α, Σ™ o^2' + ^M-i0' / ° τhus D(a) = 0, so α G C. Lemma 2.2. Every differential field k has an extension /f which is differentially closed.

Proof. Given k let / be of order n and let g be of order < n. Let f\ be an irreducible factor of / of order n. Let / = J(/ι )• Then g £ I. Let F be the fraction field of k{X}/L [Note: the quotient rule gives us a way of extending a derivation on an integral domain to its fraction field.] Let α G F be the image of X(mod /). Since / G /, /(α) = 0. Since g£I, g(ά) φ 0. Iterating this process we can build K D k a differentially closed field. The next lemma is crucial for quantifier elimination. Lemma 2.3. Let K and L be ^-saturated models of DCF. Let α G K, b G L, k = Q{ct) and / = Q(6). Suppose σ : fc —>• / is an isomorphism such that σ(cϊ) = 6. For all α G J^ there is an extension of σ to an isomorphism σ* from k(a) into L.

Proof. Let α G K. First suppose α is differentially algebraic over k. Let / be the minimal polynomial of I(a/k) , the ideal of differential polynomials in k{X} which vanish at α. Say / has order N. Let g be the image of / under σ. Let Γ(v) = {g(v) = 0} U {Λ(v) ^ 0 : A(X) G /{^} where ft has order < N}. For any ΛI, . . . , Λ n G /-PO, where each A, has order < AT, we can find /? G L such that ,0(/J) = 0 Λ ΠM/?) ^ O Thus by ω-saturation there is β in L realizing Γ(v). Extend σ by setting σ*(α) = β. It is easy to see that I(β/ΐ) is the image under σoΐl(a/k). Thus k(a) S /{/?}. If α is differentially transcendental over fc, we use ω-saturation to find β G £, /? differentially transcendental over /. We can now extend σ by sending a \—>• /?. Theorem 2.4. DCF has elimination of quantifiers.

Proof. It sufficed to show that if K, L \= DCF, k C K, k C L, a G *, 6 G K, 0(v,ΰJ) is quantifier free and K \= <£(f>,ά), then L \= 3v φ(v,a) (see [Marker] 1.5

)•

Since we may replace K and L by elementary extensions if necessary, we may without loss of generality assume that they are ^-saturated. We may also assume that k is the differential field generated by a. By lemma 2.3, we can find βeL such that k(b) S k(β). Thus L |= φ(β,a). So L μ 3v φ(v,a).

48

Corollary 2.5. DCF is complete and model complete. Proof. Let K and L be models of DCF. Then Q (with the trivial derivation) is a substructure of both fields. Every sentence φ is provably equivalent with a quantifier free sentence φ. But

Thus K = L, so DCF is complete. Every quantifier eliminable theory is model complete. Quantifier elimination leads to the following Nullstellensatz of Seidenberg. Corollary 2.6 (Differential Nullstellensatz). If k is a differential field and Σ is a finite system of differential equations and inequations over k such that Σ has a solution in some / I) fc, then Σ has a solution in any differentially closed KD k. Proof. By quantifier elimination the assertion that there is a solution to Σ is equivalent in DCF to a quantifier free formula with parameters from k. Thus if there is any differentially closed L D k containing a solution to Σ, then every differentially closed K D k contains a solution to Σ. But if there is any differential field / D k containing a solution to Σ , then by lemma 2.2 there is a differentially closed L D /. Thus Σ has a solution in any differentially closed K D k. Exercise. Let K be differentially closed. Let Σ be any set of differential polynomials mX1...Xn. Let V(Σ) = {x G Kn : f ( x ) = 0 for all / G Σ} and let I(V) = {ge K{Xι ...*„}: g(x) = 0 for all x G V(Σ)}. Then I(V(Σ)) = {Σ} the smallest radical differential ideal containing Σ. Let's make the quantifier elimination explicit. The atomic formulas in £ are of the form f(Έ) = 0 where / is a differential polynomial. Thus by quantifier elimination every formula φ(v^'a) over a differential field k is equivalent to one of the form: n

TO,

rt

k=l

where fij,9ij G k{X). Of course Λ^»,j( π ) ί ° if and only if Thus every formula is equivalent to one of the form:

49

We next show that there is an intimate relationship between types for DCF and differential prime ideals. Let k be a differential field and let SΊ(fc) be the 1-types of DCF with parameters from k. For each 1-type p(v) £ S\(k}. Let Ip = {/ £ k{X}: "f(v) = 0" £ p}. It is easy to see that Ip is a prime differential ideal. Lemma 2.7. p H+ Ip is a bijection from Sι(k) to the space of prime differential ideals over k{X}. Proof. Suppose p, q £ SΊ(fc) and p ^ q. Then there is a formula <^(v, cz) £ p\ q. By quantifier elimination there are differential polynomials f i j , g i such that φ(v,a) O V(Λ /«'» = ° Λ ΛM * °] Thus <£(v,ά) € p if and only if for some i all /y £ 7P but Ip is one to one. For any differential ideal 7, let K be a differentially closed field containing the fraction field of k{X}/I. Let p be the type over k realized by the image of the indeterminate X. Then Ip — 7, so p ι—> 7p is onto. For p £ Sι(k) we let flD(p) = RD(IP). Let K D k. If α £ if, / £ fc{X}, and /(α) = 0, we say that a is a generic solution of /, if and only if for all g £ k{X} if g
50

We say that M is atomic over A if and only if every m G M realizes an isolated type in Sn(A). Theorem 2.9 Let T be an ω-stable theory. a) (Morley) For any A a substructure of a model of T, there is M |= T, such that M D A and M is prime and atomic over A. b) (Shelah) If M and TV are prime over A, then there is an isomorphism σ : M —» TV, which is the identity on A. Corollary 2.10. If fc is a differential field then k has a differential closure K. If K and L are two differential closures of fc, then there is an isomorphism σ : K = L such that σ is the identity on k. Moreover K is atomic over k. Excercise. a) Show that a type p G SΊ(fc) is isolated if and only if there is g of order less than RD(IP) such that Ip is the only prime differential ideal containing the minimal polynomial of Ip and not containing g. b) Show directly that the isolated types are dense, [hint: For <£(υ,α), let p G S\(k) be such that RD(p) is minimal. Let / be the minimal polynomial of Ip. Show that φ(v,~a) Λ f ( v ) = 0 isolates p.] [Note: The above arguments can be used to show that DCFP has prime models even though DCFp is not ω-stable.] The following lemma will be useful when we begin differential Galois theory. Lemma 2.11. Let fc be a differential field and let K be the differential closure of k. Then CK is algebraic over Ck- In particular, if Ck is algebraically closed, Proof. Let a G CK- Since K is atomic over &, p — t(a/k) is atomic. Clearly «D(v] = 0" G p. Thus RD(p) < 1. If RD(p) = 1, then, by the excercise above, there must be f ( X ) of order 0 (ie. / G k[X]) such that "I>(υ) = 0 Λ f ( v ) / 0" isolates p. But there are c G Ck such that /(c) / 0 so this is impossible. Thus RD(p) = 0. Thus there is f ( X ) G k[X] such that /(α) = 0. We claim that α is strongly algebraic over Ck We may assume that / is the minimal polynomial of α over k. Thus f ( X ) = Σ"=Q biX\ where bn = 1. Since /(α) = 0, D(f(ά)) = 0. But n— 1

n

I>(/(α)) = LWΣ^ + ^'+^ + Σ^K i=0

»=0

Since D(α) = 0, D ( f ( a ) ) ==Σ?=o £(*•>'•Since δ« = 1. Σ?=o £(*•>' = 0. Since / is the minimal polynomial of α over fc, we must have all D(b{) — 0. Thus all of the b% G Ck - So α is strongly algebraic over Ck . The next lemma is another useful consequence of the fact that the differential closure of K is atomic over K.

51 Lemma 2.12. Let K be a differential field. Every element of the differential closure of K is differentially algebraic over K. Proof. Suppose a is in the differential closure F of K and α is differentially transcendental over K. Let ψ(v) isolate tp(a/K). Since a satisfies no differential polynomial equations over K, φ(υ) we can assume that ψ(v) is "f(v) φ 0" for some / £ K{X}. Suppose / has order n. There is b £ F such that j(n+ι) — o Λ /(&) φ 0. Clearly a and 6 have different types over k} contradicting the fact that φ isolates the type of α over K. Definition. A type p(u) £ S(A) is definable if for each formula φ(y,w) there is a formula dφ(w) with parameters from A such that for all α £ A φ(v,a) £ p if and only if dφ(a). In a stable theory all types are definable. This has a very simple proof for differentially closed fields. Exercise. Let k be a differential field and let p £ Sn(K). Show that p is definable. [Hint: Use the Ritt basis theorem to find /ι,...,/ m £ h{X} such that {g : "g(y] = 0" G_p} is the smallest radical differential ideal containing fιL - . , Λn For A e k{X}, if 0(ΰ) is «g(v) = 0" , then dφ is Vz (Λ Λ(t ) = 0 -> p(ϊj) = 0). Use quantifier elimination to get definitions of all formulas.] We conclude this section by proving that differentially closed fields satisfy uniform bounding. Theorem 2.13. Let K \= DCF. Suppose <^(a:ι, ... ,a? m ,t/ι, . . . , ? / / ) is an Cformula then there is an N such that for any ~a G K1 if {x : φ(x,ά)} is finite then it has cardinality at most N. Proof. We first note that it suffices to prove this for m = 1. If we can find uniform bounds for ψι,...,^ n , then we can find uniform bounds for \J φi. Thus by quantifier elimination it suffices to consider φ(x,y) = /\Λ(*,y) - O Λ g(x,y) + 0. Let ψ(x,y,v) be the formula

Suppose α G K1 and

52

Then the collection of formulas {φ(x, α, ί>ι),..., ψ(x,a, bn)} is inconsistent, while every proper subset is consistent. This shows that theorem 2.13 is a consequence of the following lemma. Lemma 2.14. Let K be a differentially closed field. Let Λ , . . . , fn G K{X, Y} and let φ(x>y) be the formula /\/, (^,y) = 0. There is a number s such that if {φ(x, C i ) , . . . , <^(af, cm)} is inconsistent, then some subset of size at most s is inconsistent. Proof. Let m i , . . . , r r i M be a listing of monomials in X^ containing all monomials occuring in any // (in particular assume mi = 1). Thus we can find α, j α m differential polynomials in Y such that /,- = Σ?=ι »,j r Suppose N > M + 2. We will show that for any c i , . . . ,c#, if {<^(ϊ7, Cj) : z 1 £ ^ TV} is inconsistent, then there is a subset of size M + 1 which is inconsistent. Consider F{(Z\,..., ZM, y) = Σ α »,j^j For each c; , let σj be the system of linear equations n

and let

N

j=ι Using elementary linear algebra we see that if Σ is inconsistent, there are such that Λj=ι σ «j ^s inconsistent. In this case surely inconsistent. On the other hand if Σ is consistent there are ή , . . . , ZM such that the solutions to Σ are exactly the solutions to /\J=1σ,J . Suppose {<£(«, CiJ : j — 1, . . . , M} is consistent. Let α be a solution. Building up monomials from Έ we get (1, /?2 , . . . , βu] a solution to /\^=l σij . But then (1, /%, . . . , /?M) is a solution to Σ and α is a solution to Λj=ι Φfii'cj)Thus if every M + 1 element subset of {φ(x, Cj) : j = 1, . . . , N} is consistent then the entire set is consistent. Lemma 2.14 is a special case of a more general fact. Definition. Let T be a first order theory. We say that φ(x,~a) has the finite cover property if for arbitrarily large N there are αi, . . . , a^ such that {φ(x, αt ) : i < N} is inconsistent with T while every subset of size N — 1 is consistent. We say that T has the finite cover property (FCP) if there is a formula with the finite cover property. Otherwise T is said to be NFCP. In T is unstable then T has the finite cover property. Both uniform bounding and lemma 2.14 are weak forms of NFCP. Poizat showed using the method of

53

pairs that the theory of differentially closed fields has NFCP. In fact, by a result of Shelah, if T has uniform bounding and elimination of imaginaries (which we will prove for DCF in the next chapter), then T has NFCP.

References The first work on the model theory of differentially closed fields was done by Robinson, though this work was influenced by earlier work of Seidenberg. Blum (see [Blum]) considerably simplified Robinson's axioms and was the first to use stability theoretic methods. The proof of uniform bounding given here is due to van den Dries and works equally well for separably closed fields. The model theoretic results of Morley and Shelah can be found in Sacks' Saturated Model Theory or Lascar's Stability in Model Theory. Differentially closed fields of prime characteristic are also interesting. They have a stable non-superstable theory and we can show existence and uniqueness of differential closures. See [Wood] for more information on DCFp.

§3. Elimination of Imaginaries Shelah introduced the structure Meq obtained by adding imaginary elements which are names for equivalence classes of 0-definable equivalence relations. Imaginaries smooth out many arguments from stability theory. In some cases we can show that the introduction of imaginary elements is unnecessary. Elimination of imaginaries turns out to be one of the central ideas in the model theory of fields. In particular if we can eliminate imaginaries then we may represent definable quotients as definable objects. We will show that differentially closed fields have elimination of imaginaries. We first work in a general setting. Definition. Let T be any theory and let M be a suitably saturated model of T. Let p be a (possibly incomplete) type over M. We say that B is a canonical base for p if B is definably closed and whenever σ is an automorphism of M, σ fixes the realizations of p (setwise) if and only if it fixes B pointwise. Since we do not require p to be complete it makes sense to talk about canonical bases for formulas. Lemma 3.1. Suppose B is a canonical base for φ(v, a). Then there is a formula ψtyiW) and 6 £ B such that φ(v^) *-+ φ(v^b) and $(y,T)) ψ+ ψ(y,b ) for all

54

Proof. Let Γ(ϊ7) = {φ(U) : ψ has parameters from B and φ(v,ά) —»• ψfi)}. We will show that Γ(U) —> φ(v,'a). Suppose not. Then by saturation there is c E M such that Γ(c) and -^(c, α). Ift(l?/B) — t(c/B), then there is an automorphism of M fixing B and sending c to c'. Since any automorphism which fixes B normalizes <^(ΐ7,α), we have -><£(c',α). Thus t(c/B) -+ -ιφ(v,~a). Hence there is a formula Θ(U) with parameters from B such that θ(v) £ t(c/B) and 0(U) —>• -u£(ϊ7, α). But then -ι0(ΐ7) £ Γ, contradictingj^c). Thus Γ(ϊΓ) —»• <^(U,α) and by compactness there is a formula VΌ(^, b) with 6 E 5 such that 0(ϊ7, α) <-»• VΌ(^, 6). (Here we have just reproven the well known fact that a set X is definable from A if and only if in any saturated enough model every automorphism that fixes A, normalizes

*)•

If b and 6 realize the same type over the empty set then there is an automorphism σ taking b to b . Since this automorphism does not fix 5, it does not normalize φ(v, a). Thus φ^(v^K) »&) Thus there is θ(w) 6 *(6), such that θ(c)Λcίb^(fo(v,b)^(ψQ(v,c)). Let ψtytw) = ψo(lϊ,w) Λ ί(t/J). In particular the canonical base for a formula will be the definable closure of a finite set. Definition. A theory T admits elimination of imaginaries if every formula <^(U, α) has a canonical base. The next lemma gives the connection between elimination of imaginaries and equivalence relations. Lemma 3.2. Suppose T admits elimination of imaginaries and has two constant symbols. Let M [= T and let E be a 0-definable equivalence relation on M n . There is a 0-definable / : Mn -> Mm such that xEy & f ( x ) = f ( y ) . Proof. By elimination of imaginaries and 3.1, for each formula <^(v",ά), there is a formula ψά{'v,'w) and a unique 6 such that <^(ΰ",α) «-»• ^("v, 6). By compactness we can find ψι,^.., ^n such that for all α there is an i and a unique 6 such that φ(V)~a) «-» ^<(17,6). By the usual coding tricks we can reduce to a single formula V> (a sequence made up of the distinguished constants is added to the witnesses 6 to code up the least i such that ψi works). To prove the lemma let φ(v,w) be UEw and let / be the functions Ίϊ i—» 6, where 6 is unique such that vEά O> ψ(Έ,])). The next lemma gives a test for elimination of imaginaries. We say that B is a canonical base for a finite set of types if and only if an automorphism permutes the types if and only if it fixes B.

55

Lemma 3.3. Let T be an ω-stable theory and let M |= T be suitably saturated. If every finite set of conjugate complete types over M has a canonical base, then T admits elimination of imaginaries. Proof. For any formula <^(x,y), let Eφ(y^) <& Vx (φ(x,y) «-+ φ(x,Ί}). An automorphism of M fixes φ(x, α) if and only if it preserves the E^-class of ~a. Let Pi, . . . ,pn be the global types of maximal rank that contain Eφ(y,a). We can partition {pi, . . . ,pn} into finitely many conjugacy classes. For each class we can find a canonical base B. Let A be the union of the canonical bases. Clearly an automorphism permutes the pi if and only if it fixes A. An automorphism of M permutes p\ , . . . , pn if and only if it fixes the Eφ class of α. Thus A is a canonical base for φ(x,a). Elimination of imaginaries for algebraically closed fields, differentially closed fields and separably closed fields can be proved using the following classical theorem from algebraic geometry. Definition. Let K be a field and let / be an ideal in ίίpf]. We say that k is a field of definition for / if I is generated by polynomials in k[X]. Theorem 3.4. Every ideal I in K[X] has a unique smallest field of definition k. Any automorphism of K which fixes I fixes k pointwise. Proof. Let M be a basis of monomials for K[X]/I as a vector space over K. Each monomial u £ K[X] can be written as ^α^m,- -I- gu where a u>l £ K, mi £ M and gu £ /. Let k be the subfield of K generated by all the auj. For any / £ .K^-X"], / can be written as £^ 6 u w, where each u is a monomial. Thus

=

bu(u -

If / is in /, then, since each u — Σ α U)l m, is in I and the mt are a basis for K\X]/I, each of the a — 0. Thus the u - ^α^ra,- generate the ideal /, but u — Σ α u > t m, £ k[X]. So k is a field of definition for /. _ Suppose / is a second field of definition for /. Let /i, . . . , fs £ l[X] generate /. For each monomial tί, there are gUtι , . . . , gUjS in K\X] such that iί— ^ αιt,«m* — Σ,9u,ifi Viewing the α u>l and 0 U) i as variables, we get a system of linear equations over l[X] This system has a solution in K and hence in /. But then the mi form a basis for K\X]/I, so if u - ^cU)t mt £ / we must have c Ujt = α U j t . Thus fc C /. Let α be an automorphism of K fixing 7. For each monomial u, a(u α Σ «,*'m«') =u- Σα(α",*')m«' ^ 7. Again since the mβ form a basis for K[X]/I, we must have a(aU)i) = α U j l . Thus α fixes fc.

56

Corollary 3.5. Let {/ι,...,/ n } be a set of conjugate prime ideals in . . There is a subfield k such that if α is an automorphism of K, a permutes /!,...,/„ if and only if a fixes k pointwise. Proof. Let 7 = Π^j Since the Ij are conjugate, this is an irredundant primary decomposition of I. Let k be the field of definition of I. Any automorphism of K which permutes the Ij fixes / and hence fixes k pointwise. On the other hand, if a fixes k pointwise, α fixes I. Hence by the uniqueness of primary decomposition £there is a unique way to write a radical ideal as an intersection of prime ideals), a must permute the Ij. We next give a version for differential fields. Corollary 3.6. Let {/i, . . . ,/ n } be a set of conjugate differential prime ideals of K{Xι, . . . , Xm}. There is a subfield k C K such that an automorphism of K permutes the ideal Ij if and only if it fixes k pointwise. Proof. Let J = Π^j J is a radical differential ideal. Thus by the Ritt basis theorem it is the radical of a finitely generated differential ideal. Let /i, . . . , Λ be such that J = {/i, ...,/,}. There is an N such that all /,• G K[X^ : i < mj < N]. Let J0 = J Π K[X\j) : i < m, j < N]. Let k be the field of definition of JQ Clearly any automorphism of K fixes J if and only if it fixes Jo if and only if it fixes k pointwise. By theorem 1.19 and the uniqueness of the decomposition for radical ideals, an automorphism fixes J if and only if it permutes the Ij. Theorem 3.7. The theory of algebraically closed fields and the theory of differentially closed fields admit elimination of imaginaries. Proof. a) algebraically closed fields: Let K be algebraically closed. For p £ Sn(K), let , . . . , Xn] : «/(ΰ) = 0" € p}.

This map is a bijection between n-types and prime ideals in -Kpf]. If pi, . . . ,pn is are conjugate complete types, we get a canonical base for the set by taking the field of definition for Ipl , . . . , Ipn given by 3.5. By lemma 3.3, the theory has elimination of imaginaries. b) differentially closed field: Similar using 3.6.

57

References All of the material on elimination of imaginaries is due to Poizat. It was first proved in [Poizat 3], though our treatment here more closely follows that in Cours de Theorie des Modeles. A different proof of elimination of imaginaries for algebraically closed fields is given in §4 of [Marker]. The proof given here on the existence of fields of definition for ideals is from Lang's Introduction to Algebraic Geometry.

§4. Linear Differential Equations In this short section we will review some of the basic theory of linear differential equations. This will be used in our analysis of ranks in §5. Let k be a differential field. Definition: We define the Wronskian of ΛΌ, . . . , Xn to be the determinant Xn

W(X0,...,Xn)

= χ(n) ΛQ

χ(n) Λl

Lemma 4.1: Let XQ,...,xn £ fc, then W(XQ, ... ,x n ) = 0 if and only if XQ, .. , ) X n are linearly dependent over Ck> proof: (<=) Suppose CQ, . . . , cn £ Ck are not all zero and J^ c, x, = 0. Taking the derivative: 0 = D(^c, a?t ) = 5^ct aj( . Continuing we see that = 0.

Since the columns of the matrix are linearly dependent, W(x$,..., xn) = 0. (=>) We proceed by induction on n. Suppose W(XQ, . . . , xn) = 0, then there are αt £ fc, not all zero such that = 0.

58

Without loss of generality we assume that αo = 1. By induction we may assume that W(xι,. . . , xn) ± 0. Thus a#) + £?=1 αt xp } = 0 for each j < n. Taking the derivative we see that *?> = <).

Thus

/

*<

\

=°

But then the columns of the Wronskian determinant for #ι, . . . , xn are linearly dependent unless all D(α, ) = 0. Let L(X) = X^ + ΣΓJo1 αi-X W, where α 0 ϊ . . . , α n _ι £ k. We consider first the homogeneous linear equation L(X) = 0 Lemma 4.2: If xQ, . . . , xn £ fc are solutions of L(X) = 0, then XQ,...,xn are linearly dependent over Ck proof:

W(x0...xn)=

,

.

"

=0,

as the rows are linearly dependent over k. Let K D k be differentially closed. Lemma 4.3: In K there are x\,..., xn linearly independent solutions to L(X) = 0. proof: Given a?ι,..., arm with m < n. We can find xm+ι G -K" such that L(xm+ι) = 0 but W(#ι,..., xm+ι) φ 0. (VF(a?ι,..., #m+ι) has order m so this system can be solved in any differentially closed field.) It is also easy to see that if a?ι,... ,x n are solutions to L(X) = 0. Then dXi) = 0 for any constants c i , . . . , cn. Summarizing: Theorem 4.4: If X D fc is differentially closed then there are x i , . . . , xn £ K which are linearly independent over CK such that the solution set for L(X) — 0 is exactly the span of x i , . . . , xn over Cκ We call {x\... xn} a fundamental system of solutions to L(X) = 0.

59

If 6 G K and y 0 ,2/ι are solutions to L(X) — 6, then L(y0 - yι) = L(yι) = 0. Thus if y is a fixed solution to L(X) = 6, then every other solution is of the form x + y where x is a solution to L(X) = 0. In particular if x\,..., xn £ c x :c K is a fundamental system of solutions to L(X) = 0 then {y + Σ * * » ^ ^) is the set of solutions to L(X) = b m K. Definition: Let K/k be differential fields. We say that K is a Picard-Vessiot extension of k if there is a linear differential equation L(X) = 0 and {x\,..., xn} C K a fundamental system of solutions such that K = & { z ι , . . . , xn) and Ck = Cκ We say that K/k is a Picard-Vessiot extension for L. The following theorem of Kolchin follows easily from the construction of differential closures. Theorem 4.5: Let k be a differential field with Ck algebraically closed and let L(X) = 0 be a homogeneous linear differential equation over k. There is K/k a Picard-Vessiot extension for L. Moreover K is unique. proof: Let F be the differential closure of k. By lemma 2.13 CF = C*. By theorem 4.4 we can find x\,..., xn £ F a fundamental system of solutions for L(X) = 0. Thus K = k(xι,..., xn) is a Picard-Vessiot extension of k. Suppose K\ is a second Picard-Vessiot extension of k. Let F\ be the differential closure of K\. By lemma 2.12 Cpi — Cκλ = Cfc. Since F is the differential closure of fc, we can embedded F in ί\. Let 2/1 - - 2/n be a fundamental system of solutions of L(X) = 0 such that KI = k(y\ - - 2/n) But then each x, is in the span of ( y i , . . . , yn) over Cfc and each yi is in the span of (x\ ...xn) over C*. Thus A' = KI. Thus Z/(X) = 0 determines a unique Picard-Vessiot extension of k. References Most of the material in this section can be found in any basic differential equations text (for example [Hirsh-Smale]). Kolchin's theorem on Picard-Vessiot extensions is in [Kolchin 1].

60

§5. Types and Ranks in Differentially Closed Fields Throughout this section we work inside K a very saturated differentially closed field. Recall that a is algebraic over a set B if and only if there is a formula φ(v, w) and 6 G B such that φ(a,b) and {x : φ(x,b)} is finite. We say that a is strongly algebraic over B if a is a zero of an ordinary polynomial with coefficients in the subfield generated by B. For any 6, D(b) is algebraic over 6 but not necessarily strongly algebraic over 6. The next lemma sorts out the relationship between these notions. Lemma 5.1: Let k be the differential field generated by B. Then α is algebraic over B if and only if it is strongly algebraic over k. Proof: Suppose a is algebraic over B. Consider / = I(a/k). If RD(I) = 0, then α is strongly algebraic over k. Suppose RD(I) > 1. Let f ( X ) be the minimal polynomial of /. Let K be the differential closure of k. Let fι G K{X} be an irreducible factor of /. Then f\ and / have the same order. By saturation there is 6 G K such that I(b/K) = 7(/ι). By lemma 1.10, I(b/K) Π k{X} = I. Thus 6 and α realize the same type over k. But 6 ^ K, while, since α is algebraic over fc, anything with the same type over k must be in the differential closure K, a contradiction. The other direction is obvious. Exercise. As a corollary show that the definable closure of B C K \= DCF is just the differential field generated by B. We next give a concrete algebraic characterization of forking for one types. Suppose K C L, q G Sι(K), p G Sι(L) and q C p. We will show that p forks over K if and only if RD(p) < RD(q). We begin by recalling some basic facts and definitions from stability theory. [Alternatively, the reader could just take this as the definition of forking.] Definition: Let p G Sι(k). We say that φ(v,w) is represented in p if and only if for some α G fc, φ(v^ά) G p. We say that q D p is an heir of p if every formula represented in q is represented in p. If K \= DCF and L D #, then any p G Sι(K) has a unique heir in Sι(L). We use the following as our definition of forking. Definition: Let k C /, p G Sι(i), q G Sι(/) and p C q. We say that q does not fork over k if for all M,N \= DCF such that fc C M, M U / C TV, there is Pi G SΊ(M), #ι G Sι(N) such that p C pi, q C gx and #ι is the heir of p\. We will also use the following lemma.

61

Lemma 5.2: Let fc, /,p, q be as above. Suppose for every K \= DCF with / C K there is pi £ Sι(K) such that Pi I> p and for all qι £ 5ι(^), if q\ 3 g, then Thus 5fι = α/i for some α £ L. Without loss of generality, we may assume that g\ — f\. Let φ(v^~a) be a formula in q\. By quantifier elimination, there is a differential polynomial Λ(v,α) of order < RD(q\) such that DCF \ - ( f l ( v ) = QΛh(v,w)ΪQ)-+φ(v,:w). But /ι(ϋ) = 0 £ pi and h(vj)) φ 0 £ pi for any 6. Thus φ(v,T>) is represented in Pi. Thus gi is the heir of pi and q is a nonforking extension of p. Exercise: For n-types we can give the following characterization of forking. For p £ Sn(K). Let K be a differentially closed field, p(x) £ Sn(K), and Jb C K. Then p does not fork over k if and only if V(Ip\k) is an irreducible component of V(/p), where 7(7) - {x : f ( x ) = 0 for all / £ /}. We now define several notions of rank. a) U-rank: Let p £ Sι(fc). We say RU(p) > a + I if and only if there is q a forking extension of p with RU(q) > a. For β a limit ordinal Λί7(p) > β if and only if for all a < /?, Λ(7(p) > α. In particular RU(p) = 0 if and only if p is algebraic.

62

b) Morley rank: Let p e Sι(k). For β a limit ordinal RM(p) > β if and only if for all α < /?, RM(p) > a. We say RM(p) > α+ 1 if and only if for any K D fc, if # |= DCF, then p is a limit point of the types g £ S\(K] with RM(q) > a. If # is a forking extension of p then RM(q) < RM(p). RU(p).

Thus RM(p) >

c) depth: For P a differential prime ideal in k{X}} let the depth of P be the largest N such that there are differential prime ideals P C PI C PI ... C PN We can define RH(p) to be the supremum of the depths of differential prime ideals P C K{X] where K D k and P Π i{X} = V f Note in [Poizat 2] there are three possibly inequivalent definitions of depth. Poizat refers to this notion as "height" though we find "depth" more descriptive.] Lemma 5.4 Let p £ Sι(Ar). Then RU(p) < RM(p) < RH(p) < RD(p). Proof:

1) We always have RU(p) < RM(p). 2) We claim that for any differentially closed field K the depth of a differential prime ideal is at most RD(p). Suppose PQ C PI are differential prime ideals. Let /t be the minimal polynomial of P,. If the order of fι is equal to the order of /o then /i divides /Q. Since PQ is prime this contradicts the fact that /o is the minimal polynomial of PQ. 3) We claim that RM(p) < RH(p). It suffices to prove this for types over a suitably saturated K \= DCF. In this case RM(p) > α + 1 if and only if p is a limit point of the types of Morley rank at least α. Let Dn(K) be the types of rank at least n. By induction, if p € Dn(K], then RH(p) > n. Suppose p £ Dn(K) and Ip has depth n. Let / be the minimal polynomial of Ip and let s be the separant of /. Suppose q £ Dn(K) and "/(«) = 0 Λ s(υ) ± 0" £ q. Then Iq^.Ip. Since, Ip has depth n and Iq has depth at least n, we must have p = q. Thus "f(v) — 0 Λ s(v) φ 0" isolates p in n D (K) so ΛM(p) = n. Note in particular that p is an algebraic type if and only if RU(p) = RM(p) - RH(p) = RD(p) = 0. This yields a simple but useful corollary. Corollary 5.5: If RD(p) = 1, then RU(p) = ΛM(p) = RH(p) = 1. In algebraically closed fields there are analogous notions of rank: U-rank, Morley rank, depth and Krull dimension (transcendence degree) and these notions are all equal. We next argue that the constant field of a differentially closed field is a pure algebraically closed field.

63

Lemma 5.6 Let K be a differentially closed field. Suppose A C Oχ is Kdefinable. Then A is definable in the pure field (£#,+, •)• Proof: By quantifier elimination, it suffices to prove this for sets^of the form f ( x ) = 0, where / G K{X}. Say f ( X ) = g(X) + Λ(T), where g(X) G K\X], h(X) G K{X} and every monomial in h involves some Xf' where j > 1. Thus for ~x G Cj£, Λ(af) = 0. Thus without loss of generality the definable set A is just the points in CK which are solutions to a polynomial equation over K. By definability of types (in the theory of algebraically closed fields), if B C n K is definable in the pure field K, then C^ Γ) B is definable in the pure field Cκ Thus our set A is definable in the pure field CKCorollary 5.7: If p G Sn(K) is a type of an n-tuple of constants, then RU(p) is equal to the transcendence degree of K(oi)/K where ~cϊ realizes p. Corollary 5.8: If p is the type of a generic solution of an n t h order linear differential equation L(X) = 0, then RD(p) = RU(p) = n. Proof: Let RD(p) = n. Let K [= DCF with L(X) G K{X}. Let xlt... ,zn G K be a fundamental system of solutions for L(X) = 0. There is a definable bijection between solutions to L(X) = 0 and Cg . Thus the rank of the set of solutions is equal to the rank of Cn. But RU(Cn) is the same as the rank computed in the pure algebraically closed field. For a generic solution, c are algebraically independent, thus RU(p) = n. Corollary 5.9. If p is the type of a differential transcendental, then RU(p) — RD(p) =ω. Proof. For each n, p has a forking extension where for some new element α we look at the generic solution of X^ = a. This is a type of U-rank n. Corollary 5.10 DCF has Morley rank ω + 1. We next give two bad examples. The first shows that it is possible to have RM(p) = I with RH(p) = 2. In the second we show that it is possible to have RH(p) = I and RD(p) = 2. Open Problem. Do we always have RM(p) = RU(p}Ί We first give a non-linear example where ί7-rank is equal to the differential rank. Let f ( X ) G C[X] be a polynomial with constant coefficients and consider the differential equation X" = X'f(X). Let g(X) G C[X] be a primitive of /, that is -jfe = f. Let K be a differentially closed field and let p be the type of a generic solution of X" = X'f. Suppose F D K and let c G CF - CK- Let q

64

be the type of a generic solution to X' = g ( X ) + c. It is easy to see that q is a forking extension of p and RD(q) = I . Thus RU(p) = RD(p) = 2. Consider next the differential equation X" = ^-. If we apply the same ideas we are tempted to say that for c a new constant any solution to X' = ln(X) + c is a solution to the original equation. This does not work since the second equation is not an algebraic differential equation and hence does not make sense over an arbitrary differential field. We will see that in fact the type of a generic solution to X" = ~- has Morley rank one. We first argue that this type has depth two. Let P0 be the ideal I(XX"-X') 1 1 and let Pl = I(X'). Clearly if X = 0, X" = 0. So XX" - X = 0. Thus PO C PI. Further for any constant c, I(X — c) D PI , thus P0 has depth at least two. But the depth of P0 is bounded above by RD(PΌ) — 2. Thus PQ has depth two. Lemma 5.12 shows that I(X') is the only depth one prime ideal containing XX" — X' . Before that we give a simple lemma about differentiating polynomials. Definition. Suppose f ( X ) G K [ X ] . Let /*(X) G K^\ be the polynomial obtained by differentiating the coefficients of /. That is if f ( X ) = where mt is a monomial in the various X^ , then f*(X) = Lemma 5.11. For f ( X ) G K[X], D ( f ( X ) ) = f * ( X ) + More generally: If f ( X ) G K[X, X' ... XW], then

t=0

Proof. Let / X =

a

iχi

Then:

ί

+ X'

The general case can be proved inductively in a similar manner. Lemma 5.12. Let f ( X ) = XX" - X1 . Suppose g ( X ) is irreducible of order one and / G I(g), then X1 G I(g). Proof. Let g(X) = ΣLo fl n(^)(^ / ) n > where an G K[X], N > 0 and aN φ 0. Then, by lemma 5.11, D(g(X)) = f; <(*')" + f; ^(X')"+1 + X" Σ n«» W1-1n=0

n=0

n=0

65 Let

AW = f; naa(xr + *(Σ <(*τ + Σ ^(* n=0

n=0

n=0

Consider XD(g(X)). Substituting ^ for X" ', we see that /ι(mod {/)). Since D(g(X)) and /(X) are in 7(0), we must have /i G Since /i has order one, g must divide A . The leading term of /i is X%$-X' , while the leading term of 0 is aNX'N. Thus for some λ G K we must have Xηffi = λαjy Suppose a^ — Y^u^X* '. Then X^ - £ι6l -X'ί. Then A6 m = m6m, so λ = m. It is then easy to see that for all i < m, 6, = 0. Thus αjv = bmXm. Replacing g by ^- if necessary, we may assume that α^ = Xm. case 1. m = 0. In this case ON = 1 and /i has degree N. The coefficient of X'N in /! is

. +

- vX =ΛΓN,+

.

Consider the coefficients of (X')Q on both sides of the equation. We get that

Suppose αo φ 0. There is a largest M such that XM divides OQ. Then XM+l\a*0X, but then we must have X\(N+ ^%±X), which is impossible. Thus α0 = 0. But if α0 = 0, then X'\g. Since g is irreducible and aN = 1, X' = g, as desired. case 2. m > 0 Then /i = m(X' + u(X))g, for some u(X) G /f[X]. Considering the coef1 0 ficients of (X ) we see that Xa^ = miί(X)αo As in case one, this tells us that t either α0 = 0 or X\u(X). If α0 = 0, then as above g ( X ) = X ', contradicting m the fact that αjy = X . Thus we may assume there is v(X) G K[X] such that = Xv(X). Looking at the coefficients of X1 we see that: Λ

NCLN + AΌjv + X m

Since α^ = X , a*N = 0, thus (7A

ς^ J1 - m «JV-i + σX

Thus Xm divides X a£χ l — raαjv-i- An easy calculation shows that X \VN-I. Say α N _ι = w(X)Xm, where w € K[X]. Then m

Thus I^F = mu — 3Jr. But this is impossible since v and w are polynomials. Thus we have a contradiction. Corollary 5.13. Let p be the type of a generic solution to XX" — X1 . Then p has Morley rank one. Proof. The formula XX" = X1 Λ X' ^ 0 isolates p from all other non-algebraic types. We next consider an example of an order two equation where the depth is one. Let F be a differential field and let x £ F be such that D(x) = 1. Consider the Painleve equation X" = 6X2 -f x. Theorem 5.14 (Kolchin) If η is a solution to the Painleve equation then the transcendence degree of F(η)/F is either two or zero. Corollary 5.15 If p is the type of a generic solution to the Painleve equation, then p has depth one. Proof. If RH(p) = 2, then there is a differential prime ideal I such that Ip C I and RH(I) = 1. But if μ is a generic solution for 7, then μ satisfies the Painleve equation and the transcendence degree of F(μ)/F is one. proof of 5.14. Suppose not. Then RD(I(η/F)) is one. Let / be the minimal polynomial of I(η/F). f has order one. By lemma 5.11,

But η" - 6η2 + x. Thus 0 - D(f(η))

Thus -j^X1 + (6X2 + x)j^τ + f*(X) is in I(η/F). Thus / must divide •j^X1 + (6X2 + x} §χϊ + /*PO The next lemma shows that this is impossible. The next lemma is about polynomial rings. Lemma 5.16. Let p(X,Y) c

G F[X,Y]

)^v "^~ I7*- Then q is not divisible by p.

- F. Let q(X,Y)

= Y j% -f (&X2 +

67

Proof. Suppose p divides q. The degree (in the usual sense) of q is at most the degree of p + 1, thus q = (α + bX + cY)p. ; Let j be largest such that Y occurs in some term of p. { 1 Let i be largest such that dX Y^ is a term of p. The coefficient of XΎ^ ι +l in q is zero, while the coefficient ot X γi in (α + bX + cY)p is cd. Thus c = 0. l l Similarly the coefficient of X + γi in q is zero, while in (a + bX)p it is 6d. Thus 6 = 0. Thus for some α £ F, g = αp. Let p = Y^j=QpjX^ where PJ £ F[Y] and pn φ 0. For notational simplicity we let pi = 0 for i < 0 or i > n. Since q = αp,

This yields the system of differential equations:

We solve for j = n + 2, n + 1 and n. 6^ = 0. Thus pn = iί0, for some w 0 G F with % / 0. 6 ^y1 = 0. Thus p n _ι = VQ, for some VQ E F.

and

6 p£γ2 = — lίό + αw 0

Thus p n _2 = t^o^ + ^o» where WQ = — ^(u'Q — αιt0)

We claim that for any k we can write:

The above arguments show that it is true for fc = 0. Assume it is true k. 1= n-3k-l Using the inductive assumptions we see that = (n - 3*)«fcy2*+1 + (-v'k + avk Thus p n _ 3 *_3 = «ife+ιy2*+2 + rfc+1y2fc+1 + . . ., where =

1 fn-3k\ ~6 (2ikT2 J «*•

and Γk+1 =

~

v'k - avk - (n -

6 2* Similar arguments for the cases j = n — 3k — 2 and j = n — Zk — Z yield

and Pn-3(*+l)-2 =

where: Vk+1 =

\ ~ 6 -2ΪT2- J

(3)

l)aru>t + (n - 3fc -

and

(1) gives a recursive definition of u^. This yields:

We know that t/0 is nonzero. If n is not divisible by three, then (7) would imply that for all i, u^ φ 0. Since pn-3k = 0 for Ar > ^, we must have n = 3m for some m. Then by equation (7) for u^ = 0 if and only if k > m + 1. Using (7) and (5) we have

β This giver a recurrence relation for the wk. Solving this we get:

.

(8)

For 1 < / < 3, pn-3m-i = P-i = 0, thus um = rm = vm = sm = wm=tm = 0. Since wm = 0, equation (8) implies that r/ό — auQ = 0 (note that all of the terms in the product are 1 nonzero). But then (8) implies that for all fc, wk = 0.

(9)

and (7) implies that for all fc, ' - aUk = 0.

(10)

u k

Since all of the wk = 0, (3) implies that for all k vk — 0 and then (2) implies that all of the rk = 0. Using (7) and the fact that all of the r^ = 0 we can simplify (6) to -3i+ 3

l/n-3k-2

Solving this reccurence relation for tk we get:

=

/-l\

t+1

XU

^n-Si + S

n-Si +

°Σ(Π— — Π —2ϊ— h=0 \i=l

ί=Λ+l

/

/Ί1x

'-Λ*. Λ n - 3 i + l Since tm = 0, the above equation yields:

Substituting the above into (11) and simplifying we have that if 1 < k < m then:

In particular for each fc such that α < k < m, there is a positive rational number βk such that (-l)k+ltk = βkxuQ. Thus (-1)*^ = A(«tιό + «o) [note: this is the one point in the proof where we use the fact that D(x) = I (though any positive rational would do)]. Thus (-l)*+1(*ί - <4) = βk(x(u'0 - αtio) + no). But u'0 — auo = 0, so (— l)*+1(ί'j. — at'k) is a positive rational multiple of UQ. Since all of the u>jt = 0, (4) simplifies to: - 3* So

70

But for 1 < k < m, (—1)*^ — atk is a negative rational multiple of UQ. Using this and the fact that s 0 = 0, it is easy to show by induction that if k 1 < k < m, then (—l) Sk is a positive rational multiple of UQ. In particular sm φ 0, a contradiction. This concludes the proof. Note: The 6 in the Painleve equation plays no role in the above proof, (ie. it would work just as well for X" = X2 + x). Lemma 5.16 can also be used to show that Cp(η) — CF> References All of the details on forking and ranks can be found in Lascar's book Stability in Model Theory. Most of the material in this section is taken from [Poizat 2]. The analysis of the Painleve equation is due to Kolchin, extending work of Kovacic. As far as I know it is unpublished. The version I have seen is in a letter from Kolchin to Carol Wood.

§6. Non-minimality of differential closures In this section we will show that differential closures need not be minimal. We will find a differential field k with differential closure K such that there is a differentially closed L D k with L properly contained in K. In this case L is also a differential closure of fc, so K and L are isomorphic over k. Thus we can properly embed K into itself fixing k. This theorem was proved independently by Kolchin, Shelah and Rosenlicht. We will follow Rosenlicht's proof. The first lemma gives a criteria for telling if a prime model is minimal. Lemma 6.1. Let T be an ω-stable theory. Suppose M \= T is prime over A. If M is minimal over A, then whenever 7 C M is a set of indiscernibles over A, / is finite.

proof. Suppose M is minimal over A and / C M is an infinite set of indiscernibles over A. Let 6 E 7 and let J = 7 \ {b}. Let N \= T be prime over A U J. There is an elementary embedding of TV into M fixing A U J. Thus, since M is minimal over A, N = M and M is prime and atomic over A U J. There is a £ A, GI, . . . , cn £ J and a formula <^(t;,ά, c) isolating the type of 6 over A U J. Let d G J \ {ci,..., cn}. Since φ(v,a,c) isolates t(b/A U J), M |= <£(v,α, c) —> v φ d and

71 Since b and d are indiscernible over AU{CI, . . . , cn}, we must have M |= φ(d, α, c), a contradiction. The next theorem is the algebraic core of the proof. This result will also be useful in the next section when we build many models. Theorem 6.2 (Rosenlicht). Let k C K be differential fields such that the CK is algebraic over Ck Let C denote Cfc. Suppose / G C(X)} cι,...,c n G

Suppose xι,£2 £ K are solutions to Xz = αt /(Xt ), where 01,02 G fc. If xi and #2 are algebraically dependent over fc, then each x, is algebraic over k or We give two partial fraction decompositions which will prove useful.

Ex 1): f ( X ) = τfc.

/(*) ~ X

d

Ex 2): f(X) = X3- X2. Let u(X) = Zf± and v(X) = jf. Then du

-1

So

and

1

1 X - X2 1 1 3

dX'

CoroUary 6.3. Let C be a field of constants and f ( X ) = ^ or f ( X ) = X3 — X2. Let K be the differential closure of C and let xlt. .. ,xn £ K be nonconstant solutions to X'{ = α
72

proof. By 2.13 CK is algebraic over C. We first examine the case where f ( X ) = ^T In this case v(X) = X. If a,jv'(xi) — α,V(xj), then did,α 7 α, -—-— . J - = 3 1 + x{ 1 + Xj

In this case x, = Xj . Suppose c is a constant solution to X1 = α, /(X). Then /(c) = 0, so c = 0. Let #i, . . . ,~zn £ A" be nonconstant such that x\ — o>ij(xϊ) and n is minimal such that EI, . . . , xn are algebraically dependent over C. n = 1. Then #ι is algebraic over C*. But then xι is constant (by 2.1), a contradiction. n > 1. Then xn and x n _ι are algebraically dependent over G(XI, . . . ,x n -2) Neither xn-i nor xn is algebraic over C(xι, . . . , #n-2), so by Theorem 6.2, α n v(x n _ι) / = αn-iK χ n)' But then, z n _ι = z n , a contradiction. In the second case v(x) = £. Thus if αiί/(xj)αjt/(xi), x^ = x; . The only constant solutions of X' = a i f ( X ) are zero and one. The remainder of the proof is similar. Corollary 6.4. Let C be a field of constants. Let K be the differential closure of C. Then K is not minimal over C. proof. Since K is differentially closed it contains infinitely many solutions to t/ = /(y), where / is one of the above functions. Let xi, X2ι be NO nonconstant solutions. By 6.3 the x» are algebraically independent over C. For any Xjl , . . . , Xj m , since xJ - /(*,-) and /(X) G C[X], C^, . . . ,xjm) = C(x^...,xjm). Thus the type of Xjl , . . . , Xjm is determined by = /(v ) Λ ^ ^ 0) Λ ?(»!,. . . , wm) ^ 0, for p a nonzero polynomial over C. Thus the x, are a set of indiscernibles. So, by 6.1, -fiΓ is not minimal over C. The proof of Rosenlicht's theorem uses the abstract theory of differential forms. Suppose k C K are fields. We define Ωκ/k the space of differential forms on K over k (when no ambiguity arises we will drop the subscripts). Let Ω be the ^-vector space generated by the set {dx : x £ K}, where we mod out by the relations: d(x + y) = dx + dy, d(xy) = xdy + ydx, and d(a) = 0 for α G k.

73

It is easy to see that for p(X) G k[X]9 d(p(x)) - -J£(x)dx. The space of differential forms Ω satisfies a universal mapping property given by the following lemma. Lemma 6.5. If D : K -> K is a fc-derivation (ie. k C C#), then there is a ^-linear ξ : Ω -* K such that D = ξ o d. Proof. Let £(cfx) = D(x). This is well defined since: ξ(d(x + y)) - L>(* + y) = £>(*) + D(y) = ζ(dx) + ζ(dy), ξ(d(xy)) = D(xy) = x/J(y) + yD(x) = xξ(dx) + yξ(dx), and ξ(da) = D(ά) = 0= ί(0), for α G k. We next show that the dimension of Ω as a X-vector space is equal to the transcendence degree of K/k. The proof uses two facts about extensions of derivations which we summarize in the next lemma (for proofs see Lang's Algebra). Lemma 6.6. Let K be a field and let D : K —> K be a derivation. a) Let α be any element of K(X), then D extends to a derivation D* : K(X) -> K(X), with D*(X) = α. b) If L/K is separable algebraic, then D extends to a derivation on L. Lemma 6.7. dim^Ω = td(K/k). Proof. _ Suppose 1 1 , . . . , tn G K and p(X\ , . . . , Xn) G k[X] is of minimal degree such that p(ΐ) = 0. Then

i=ι Since the degree of p is minimal, for some i, j£-(t) / 0. Thus ctti, . . . ,cK n , are linearly dependent over K. Thus dimχ(Ω) < td(κ/k). Suppose ti, . . . ,tn are algebraically independent over fc. By 6.6, we can find derivations Di : K -+ K such that , Let & : Ω -> K such that A = & o d. Suppose α i , . . . , a,, G K and

otherwise.

74

Then

J=l

= α;. Thus dti, . . . , cftn are linearly independent, so dimj^Ω > td(K/k). Corollary 6.8. If t £ ff , then t is algebraic over k if and only if dt = 0. Suppose D : /f —+ /f is a derivation. Let D1 : Ω —»• Ω, be defined by

The following properties are easy to verify for x 6 K, ω, η £ Ω: D'(ω + η) = D'(ω) + D'(η)

XX(«ω) = D(ar)ω + xD'(ω) D'(dx) = d(D(x)). Lemma 6.9. Let D : K —> K, be a derivation such that D\k is a derivation on fc. If x,y in K are algebraically dependent over Ck, then D(y)dx = D(x)dy and Proof. Let p(X,Y) E Ck[X,Y], be such that p(x,y) = 0 Since p(x, y) = 0, d(p(x, y)) = 0. But Q

Λ

So

Also, since the coefficients of p are constant, D(p(x, y)) (see lemma 5.11). Thus

So £>(a;)eίί/ = Finally D'(xdy) = D(x)dy + xd(D(y)) = D(y)dx + xd(D(y)) = d(xD(y ) ).

75 Lemma 6.10. Suppose KI, . . . ,tι n , υ £ K and all the Ui are nonzero. Suppose ci, . - . , cn £ k are linearly independent over Q. Let n

j

^-Λ dtif α; = αi; + > c, - . Then α; = 0 if and only du\ = ...=. dun = dv — 0 (ie. all of the iί, and v are algebraic over k). Proof. case 1. ιtι, . . . , un are algebraic over fc. Then all of the dut = 0. Thus ω = 0 if and only if dv = 0 if and only if t; is algebraic over k. Thus we may assume that some Ui is transcendental over k. Without loss of generality assume u\ is transcendental over k. We will show this leads to a contradiction. case 2. u\ is transcendental over k and 1/2, . . . , u n j v G &(wι). We can give formal Laurent series expansions for Uj and v in terms of u^. Say 00

otjtiU\,

and

i=ί

Then

i=/-l

In particular in this expansion dv = /(ιiι)diίι, where /(«ι) is a Laurent series where the coefficient of t/j" 1 is zero. Thus —3- — duι(πijU^1 + higher degree terms) Uj

If ω = 0, then comparing the lίj"1 coefficients we see that

This is a contradiction, since c i , . . . , cn are linearly independent over Q.

76

Finally we show that we can reduce to case 2. Suppose u\ is transcendental over k. Let HI , ti . . . tm be a transcendence base for iίχ , . . . , un , v over k. Consider the natural homomorphism φ : Ω#/fc —> Ωjκγ]b(tι...t n ) If ω = 0, then <^(ω) = 0. We replace k by k(tι . . .tn). Thus we assume that HI is transcendental over k and t/2 . . ttn, v are algebraic over fc(ι«ι). We also replace K by a finite algebraic extension of k(uι , . . . , un , v) so that K/k(uι) is Galois. Let G = Gα/(ff/i(tiι)). For σ G G, let ω

=

i

\-dσv

Each ωσ = 0. Let 77 = Σσ€Gωσ. For j = 2 , . . . , n , let ιιf=

Then t/f G Ar(tiι) and

dυf =

Let

Thus , = [/C:*(«1)]c01 1ίίί. 6

Replacing iίj by w* for j > 2, v by v^ , and Ci by [K : AΓ(UI)]CI, we have reduced to case 2. Remark. The fact that the constants ci, . . . ,cn are linearly independent over Q is a red-herring. Note that: \ d(ary) _ d£ , ^ V a?y a? "•" y

ϋ) ^ - «f fom G N. Using these two facts it is easy to see that for any ^ c» v^ can ^e rewr^ten as 5^tf~^ where the 6j are linearly independent over Q. We are now ready to prove theorem 6.2 which we repeat for convenience.

77

Theorem 6.2 (Rosenlicht). Let k C K be differential fields such that the CK is algebraic over Ck Let C denote C*. Suppose / G C(X), cι,...,c n G

Suppose £ι,#2 € K are solutions to Xt = α,/(X, ), where αι,α2 G k. If #ι and X2 are algebraically dependent over fc, then each x, is algebraic over k or proof of 6.2. We may assume that K = k ( x ι , X 2 ) . Suppose x\ and x^ are algebraically dependent over fc, but neither is algebraic over k. Thus td(K/k) = 1. By lemma 6.7, A'm/fΩ = 1. In particular, -^v generates Ω as a /^-vector space. Thus there is a nonzero c G K such that

We claim that c is a constant. By lemma 6.9 (with x = T^TY, y = £»)•

since αt G fc. Thus

But then D(c) = 0 so c is constant and hence algebraic over k. We now use our expression for / and the fact that d(w(x)) = fy
78

Soby.(l)

Since c £ CK, c is algebraic over Cfc. Thus by corollary 6.8 dc = 0. Thus we can rewrite (2) as

j=ι

We now apply lemma 6.10 (and the remark following it) to (3). Thus d(v(x2) - cυ(xι)) = 0

(4).

Finally,

dx

Similarly

By (1) and (4)

Thus as desired. We conclude this section with a proof that in Rosenlicht's extensions we do not add new constants. This will be useful in the next section. Definition. We say that E/F is a function field if there is t £ E transcendental over F and E is a finite algebraic extension of F(t). If F is algebraically closed, then function fields correspond to isomorphism classes of smooth projective curves over F. If E/F is a function field, then the genus of E is the genus of the corresponding curve. Lemma 6.11. Let K/k be differential fields such that K/k is a function field and Cfc is algebraically closed. If CK φ Cfc> then Cκ/Ck is a function field and the genus of Cκ/Ck is at most the genus of K/k.

79

Proof.

Suppose CK φ Ck and t G CK — C^. Then t is transcendental over C^. The arguments from the proof of 2.13 show t is transcendental over k. claim. Ck(t) = Ck(t). Suppose D(p(t)) = 0, where p(X) = Σaiχί £ k[X]. Then D(p(t)) = D(t) Σ iαrf'-1 + Σ

D

(α ')*'' = Σ

Since t is transcendental over fc, we must have all £>(α, ) = 0, so α, G Cfc. Thus p(0 € Cfc(ί). Suppose ppQ and q(X) are in fc[X] and -D(^τ|τ) = 0. We may assume that q is monic and that for any go of lower degree there is no po such that ^W = ^W. Since £>($}) - 0, q(t)D(p(t)) - p(t)D(q(t)) = 0. But then §gg =° gjj. But if q(X) = Xn + Σni=vbiχi> then D(q(t)) = Σ^Q D(bi^^ contradicting the minimality of q. claim. Cκ/Ck is a function field. We know that t is a transcendence base for K over k. Assume that K/k(t) is an algebraic extension of degree N '. Let x G CK ~ Gfc(ί) Let /(-X") G fc(t)[X], be the minimal polynomial of x over .K. The degree of / is at most N . Let f ( X ) = Xm +ΣbiXi 0 = D(f(x)) = Σ"=o β(6.>'' Since this polynomial has lower degree, we must have all of the D(b{) = 0. So f ( X ) G C^) [X] . Thus [CK : Cffc(t)] < TV. So Ctf/Cfc is a function field. Let α be a generator for Cκ/Ck(t). Let f ( X , Y ) G CΆ ^Y] such that f ( t , Y ) is the minimal polynomial of α over Ck(t)- By the above arguments f ( t , Y ) is also the minimal polynomial of α over &(/). Thus k(t,a) is a function field of the same genus as Cκ/Ck- Since fc(/,α) C K, the genus of X/fc is at least the genus of k(t, a)/k (there can be no maps from a curve of genus g to a curve of genus g\> g [by Hurwitz formula]). Theorem 6.12. Let k be a differential field such that C = Ck is algebraically closed. Let f ( X ) G Ck(X) and let x be a solution of the differential equation D(X) = f ( X ) , where x is transcendental over fc. Suppose that γτχ\ is not of the form cj^/u or c j% for any ti or v G C(X), c G C. Then (?*(*) = C. Proof.

Suppose Ck(x) φ C. By 6.11, Ck(x) is a genus 0 function field over C. Thus there is t G Cfc^) such that Ck(x) = C(t). Consider the non-zero differentials dt and jπ in Ωk(x)/k. By 6.7 there is g G k(x) such that j4|τ = gdt. &(gdί) = D(g)dt + gD'(Λ) = D(g)dt + gd(D(t)) = D(g)dt. While by 6.9

80 Using the partial fraction decomposition of J^T G C(x), we can write

dx _ Y^ dui J\x)

u

, _ι

ί

where c, G C, Ui,v G C(x). Using the remarks after the proof of 6.10 we can choose this decomposition so that c i , . . . , cn are linearly independent over Q. Since g G C(t), we can use the partial fraction decomposition of g to write ,

m

gdt = Σh—- + dy

where 6, G C and Wi, y G Let C i , . . . , cn, c n +ι,..., CAT be a basis for the span of c i , . . . , cn, 6 1 , . . . , bm over Q. Using the remarks after 6.10, letting HJ = 1 for j = n + 1 , . . . , N and suitably defining the iϋ, , we can may assume: N

dx

Σί=ι Ci

'aui

\-dv

Uί

N t=l

where 6t = -^ for some M G Z. Note that

Me,•i

αtίi

αiί;,c

Ui

\

Wi

/ d( u^ /Wi)\ — ci (

\

TϋΓΊ Uf JW%

)

Thus we may use the fact that gat — j^ to conclude that N

M«M /-,,Λ

ί

By 6.10, d(uf /Wi) — 0 for each i and d(Mv — y) = 0. Since A:(x) is a purely transcendental extension of fc, by 6.8 each u^ JW{ G k and Mv — y £ k. For each i, D(£) = ^^-^(ti,-), since tι;, G C(ί) - Ck(x). Thus ^gil e t. We also have D(υ) G *. But t i i , . . . ,tι n ,t; G C(a?). Thus ^^ and D(v) G t Π C(x) = C. For any Λ G C(x), D(Λ) = j^D(x) = -j£f(x). At least one of w i , . . . , UN,V θut

is not in Ar, for otherwise c?ar = 0. Thus at least one of -J*-/ or |^/ is a nonzero element of C. Thus π is of one of the forms stated in the theorem.

81

References The nonminimality of differential closures was proved in [Kolchin 3], [Rosenlicht 1] and [Shelah]. Shelah's and Rosenlicht's arguments are discussed in [Gramain 1] and [Gramain 2]. [Rosenlicht 2] contains some of the theory of differential forms that we use. This work is an extension of earlier work of Ax. [Brestovski] contains several extensions of Rosenlicht's ideas.

§7. The number of non-isomorphic models In this section we will prove that if /c is uncountable, then there are 2* non-isomorphic differentially closed fields of cardinality «, while also analyzing orthogonality and strongly regular types. The number of countable models was only recently shown to be 2N° by Hrushovski and Sokolovic. Pillay's paper in this volume contains a proof of this result. [Through out this section we assume a reasonable knowledge of stability theory. References [L] are to Lascar's Stability in Model Theory, while [B] is Baldwin's Fundamentals of Stability Theory] We say that a and 6 are independent over k if the t(~a/k(b)) does not fork over k. We write άX^fc. Recall that the α X j if and only if RD(a/k) = RD(a/k(b)). We say that a type is stationary if over any extension of the domain there is there is a unique non-forking extension. For p G SΊ(fc), p is stationary if and only if the minimal polynomial of p is absolutely irreducible. Lemma 7.1. Suppose K \= DCF and F is the differential closure of K(b). If α G F-K then α ^V*/ K.6.

Proof. Let ψ(v,b) isolate t(a/K(b)). For all m G #, ψ(υ,b) -* v φ m. Suppose αX^t. By symmetry b^κa ( see [L] 3.5). Thus t(b/K(a)) is the heir oft(b/K). Since t(b/K(a)) represents ψ(v,w), there is α 0 G K such that V>(α 0 ,6). But then ψ(a0lb) —»• α 0 φ OQ, a contradiction. Note that the above argument works for any stable theory with prime models.

Definition. Let K |= DCF and p,q € Sι(K). We say that p and q are orthogonal if and only if for any a realizing p and 6 realizing g, αX 6. We write

The above notion is usually called almost orthogonality. For types over models of an ω-stable theory these notions are equivalent (see [L] 8.23). If p G S(k) and q G 5(/), we say that p _L q if and only if for any differentially

82

closed K D k U /, if p' and q1 are non-forking extensions of p and q to K, then p1 _L g'. In general if p _L g and p1 and #' are nonforking extensions of p and q respectively, then p1 J_ q1 . Lemma 7.2. If A |= DCF, p,q G 5Ί(#), p -L ?, α realizes p and ί1 is the differential closure of K(a), then q is not realized in F. Proof. Clear from 7.1. Lemma 7.3. Suppose F D K are differentially closed, φ(v) is a formula with parameters from K and every element of F that satisfies φ(x) is already in K. Let α G F — Jf , let p = t(a/K) and let # G Sι(K) be a type containing φ(v). Then p -L ςf. Proof. Let b realize q. Let r(A") G AΓ{X} be the minimal polynomial of q. If δj^α, there are g(X),h(X) G #{α){X}, such that g(b) = 0, the order of g is less than the order of r, the order of ft(^) is less than the order of g(X) and

Since φ(v) has no new solutions in F, {* G F : F |= 0(2?) = 0 Λ ή(a?) / 0} = {x G ff : F f= 0(x) = 0 Λ ft(x) ^ 0}.

By definability of types and model completeness, there is a formula ψ(v) with parameters from K such that {x G K : F \= g ( x ) = 0 Λ h(x) φ 0} = {x G A' : # μ V(«)}= {^ € F : F |= ^(x)}. Note that V>(6) holds. But F \= "there are polynomials g and ft such that g has order less than r and Λ has order less than g such that (g(x) = 0 / \ h ( x ) φ 0) if and only if ψ(x). Thus by model completeness there are #0,^0 G K{X} such that g v ( x ) — 0 Λ ΛO(#) / 0 is equivalent to ψ(x). In particular ^o(^) = 0 contradicting the fact that r is the minimal polynomial oΐt(b/K). As an application of 7.3 suppose p G Sι(K) is the type of a differential transcendental. Let Kp be the prime model over a realization of p. We first note that every element of Kp \ K is differentially transcendental over K. Suppose not. Let 6 G Kp \ K, and suppose /(&) = 0 for some f ( X ] G K{X}. Then RD(b/K) < ord(f), but by 7.1 a^Rb. Thus RD(a/K(b}} < ω. But RD is transcendence degree. Thus iΐtd(K(a,b)/K(b)) < ω and td(K(b)/K) < ω, then td(K(a)/K) < ω contradicting the fact that α is a differential transcendental. In particular if / G K{X}, f ( X ) = 0 has no solutions in Kp — K. Thus by 7.3 if q G Sι(K) and q φ p, q _L p. Definition. Let K, F \= DCF. Let p G S(F). We say p ± K if and only if for all q G S(K), if q1 is a non-forking extension of q to F then p _L q' . We use the following fact (see [B] VI 2.23).

83

Lemma 7.4. If M C N \= T and / is an elementary map with domain N such that N^Mf(N), then p _L M if and only if p _L f ( p ) . Definition. T has the dimension order property (DOP) if and only if there are MO, MI, M2, M3 models of T such that: 1) MO C M! Π M2 3) MS is prime over MI U M2 . 4) There is p such that p _L MI, p J_ M2, and p ./. MS. The interest of the dimension order property is the following theorem of Shelah (see [B] XVI). Theorem 7.5 If T is ω-stable with DOP, then for any uncountable K there are 2 Λ non-isomorphic models of T of power K. Theorem 7.6 Differentially closed fields have DOP. Proof. Let K |= DCF. Let &ι,6 2 be independent differential transcendental over K . Let Ki, i — 1, 2 be the differential closure of K(b{). Let K% be the differential closure of K(bι, 6 2 ). Let p £ SI(KS) be the type of a generic solution of X1 = 6ι& 2 /(X), where f ( X ) = X3-X2 (or f ( X ) = JL-). Clearly p / K3. We claim that p L K\. By 7.4 it suffices to show that if 63 is differentially transcendental over K, b$^ 6 2 , and q is the type of a generic solution of Let F be the differential closure of -K (61,^,63) and identify p and q with their non-forking extensions to F. Let #ι and x 2 be realizations of p and q over F. We claim that £ιX F # 2 . Let L = F{xι,x 2 ). Since zj £ F(xi), it is easy to see that F(XΪ) = F(zz ) and L = F(XI, x 2 ). Since RD(p) = RD(q) = 1, these are types of [/-rank. If t(x 2 /F(xι}) forks over F, then x 2 is algebraic over F(XI). We will apply Rosenlicht's theorem with k — F and K = L. We need to show that CL is algebraic over C^. By theorem 6.12, CF — C F ( X I ) In general if K/k is algebraic then Cκ/Ck is algebraic. [Let c £ CK, let Ϋ^aiX1 be the minimal polynomial of c over fc, where the leading α z = 1. Then 0 — D (Σaicί} = ΣD(ai)cί + D(c)ΣίaίJ~l So Σ^(αO^Z vanishes at c but this has lower degree unless all of the α^ are constants.] Thus CL is algebraic over CF. By Rosenlicht's theorem, 6ι6 2 v(x 2 ) / = 6ι63τ;(xι)/. As we saw in 6.3, this implies x\ = x%, but this is impossible since 6χ6 2 φ 6163. Similarly p _L ίί2 , so p witnesses DOP. Corollary 7.7 For « > NI, there are 2* non-isomorphic differentially closed field of power «.

84

The idea of the proof is the following. Fix M a differentially closed field of power K containing (aaιba : a < /c) independent differential transcendentals. Let R be a binary relation on K. We can find MR differentially closed of 1 power K. Such that R(a,β) if and only if X = aabβf(X) has NI solutions and 1 -ιβ(α, /?) if and only if X = aabβf(X) has NO solutions. This idea can be used Λ to build 2 non-isomorphic models. (For example this shows that if Q is the quantifier "there exists uncountably many" DCF is unstable in L(Q).) We conclude this section with some remarks on strongly regular types and orthogonality. Notation: If K [= DCF and p £ Sι(K) we let Kp denote the prime model over a realization of p and we let fp denote the minimal polynomial of p. Definition. Let K [= DCF. A nonalgebraic type p £ Sι(K) is strongly regular if and only if for any a £ Kp \K,Ίt fp(a) = 0, then p = t(a/K). If p £ Sι(k) is stationary, p is strongly regular if and only if for any differentially closed K D fc, the non-forking extension of p to K is strongly regular. If K, F \= DCF, K C F, p £ Sι(K), q £ 5Ί(F), g is a non-forking extension of p and p is strongly regular, then q is strongly regular. (See [L] 8.9). Two important types are easily seen to be strongly regular. Let tc £ Sι(K) be the type of a new constant and let tg £ Sι(K) be the type of a differential transcendental. Clearly every constant in Ktc — K realizes tc and every new element of Ktg — K realizes tg (see the argument following 7.3). Note that in the case of tg the minimal polynomial is 0. The next lemma shows that strongly regular types are abundant. Lemma 7.8. If F, K \= DCF and K C F then there is α £ F - K such that t(a/K) is strongly regular. Proof. Choose a £ F - K such that RD(a/K) is minimal. If RD(a/K) = u>, then a is differentially transcendental over K, and t(a/K) is strongly regular. Otherwise, let / be the minimal polynomial oft(a/K). If 6 £ F—K and /(δ) = 0, then RD(b/K) is at most the order of /. By the minimality of RD(a/K), RD(b/K) is equal to the order of /. Thus / is the minimal polynomial of t(b/K), and t(b/K) = t(a/K). Hence t(a/K) is strongly regular. Lemma 7.9. If K |= DCF and p £ Sι(K) has RU(p) = 1, then p is strongly regular. Proof. Let a realize p and let Kp be prime over K(a). Suppose 6 £ Kp \ K and /p(6) = 0. By 7.1 a£χb. Thus RU(a/K(b)) = 0 and α is algebraic over K(b). Let g(X) be the minimal polynomial of t(b/K). Since /(6) = 0, the order of g is at most the order of /. The order of / is equal to the transcendence degree of K(a)/K,while the order of g is equal to the transcendence degree of K(b)/K. Since a is algebraic over K(V), f and g must have the same order. But then

85 since / G /( a contradiction. Thus K' — K contains a solution d to / P (X) — 0. Since p is strongly regular t(d/K) = p. Thus K1 contains a realization of p, and hence is prime over a realization of p. By uniqueness of prime models K1 = Kp. Definition. We define the Rudin-Keisler order on Sι(K) as follows. Let p, q G Sι(/f). We say p >RK q if and only if q is realized in Kp. We say p ~RK q if and g >β# p. Corollary 7.12. If p G Sι(K) be strongly regular, q G SΊ( K') is non-algebraic and p >fl# g, then p ~βχ g. Proof. We can embed Kq C Kp such that /Cg φ K. realization of p.

By 7.10, /Cg contains a

Lemma 7.13. Let p, g, r G 5ι(/C). Suppose r >## p and r J_ g, then p _L g. Proof. Let α, 6 realize p, q. Since r >## p, we can find d realizing r such that α is in the differential closure of K(d). Since q J_ r, &X χ d In particular we can find a differentially closed field F D K(d) such that t(o/F) is the heir of the g. Since K(a) C F, &X#α τhus p i g . Corollary 7.14. Let p,g G Sι(K) be strongly regular. equivalent: i) #„ S /fg ϋ) P >RK q ϋi) P ~RK q iv) p /. q. Proof. i) => ϋ) => iϋ) => i) is clear from 7.11,7.12.

The following are

86

π) :z> iv). Is clear from 7.2. iv) => ii). Suppose p ^RK ΛΛΓ p, thus by 7.13 p _L g, as desired. Strongly regular types are important because they can be assigned dimensions. (The reader is referred to [B] chapter XII for details.) Let k C K.

We say that A C K is k-free if and only if for all a £ A

αl^-H If p £ SΊ(fc), we say that B C K is a p-base for /f if it is a maximal Ar-free set of realizations of p. If p is strongly regular, then J^ is transitive on the realizations of p. Thus any two p-bases have the same cardinality. We call this cardinality the p-dimension of K/k. We denote this as diπι(p\K). If fco5 &ι are finitely generated, pi £ SΊ(fcj) is strongly regular, and K D fco U fci, then dim(pQ\K) differs from dim(pι]K) by at most a finite amount. Two dimensions are clearly important invariants of a differentially closed field. Let tc £ SΊ(Q) be the type of a new constant and let tq £ SΊ(Q) be the type of a new transcendental. For any differentially closed field K, let IC(K) = dim(tc\ K} and Ig(K) = dim(tg\ K). It is easy to see that for pair of cardinals κ,λ, there is a differentially closed field K with IC(K) = κ> and Ig(K) = λ. Until the work of Hrushovski and Sokolovic the only types known that were nonorthogonal to tc and tg were trivial types like those arising from Rosenlicht's examples. This lead Lascar to conjecture that perhaps any strongly regular type which is orthogonal to tc and tg is No-categorical. Lascar's conjecture would have implied that the number of countable models is NO- Indeed a countable model would be determined up to isomorphism by Ig(K) and IC(K)> The work of Hrushovski and Sokolovic shows that this is far from true. There are many locally modular strongly regular types which are not No-categorical. These matters are discussed extensively in Pillay's article in this volume.

References Shelah ([Shelah 1]) proved that in uncountable cardinals DCF has the maximal possible number of models. The analysis of orthogonality and strongly regular types is from [Lascar 1].

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§8. Differential Galois Theory Let K/k be differential fields. We define G(K/k) the Differential Galois Group of K over fc, to be the group of differential automorphisms of K which fix k point wise. We begin by looking at some important examples. Examples: 1) Adjoining an integral: Let a £ k. Consider the equation X' = α. Let it be a generic solution of 1 X' = a over k and let K = k(u). Since u = a £ k, K = k(u). If σ £ G(K/k), then cτ(ιt)' = α, thus for some c £ CK> u + c determined a differential automorphsim of K fixing k. We will assume that C* is algebraically closed. Then by theorem 4.5 K/k is Picard-Vessiot (the equation X" — *-X' = 0 has linear independent solutions 1 and it). Indeed if X1 = α has no solution in fc, then K/k is Picard-Vessiot (see [Kaplansky]). Since CK — Cfcj the above argument shows that G(K/k) is isomorphic to the additive group of Ck 2) Exponentials Let a £ k. Let it be a generic solution of X' = aX over k. Let K = k(u) = fc(ιt). Suppose CK = Ck (for example, suppose Ck is algebraically closed), then K/k is Picard-Vessiot. If σ £ G(K/k), then σ(t/) = cit, for some c £ Ck Moreover if c £ Cfc, then u ι—> cit determines an automorphism of Jϊ. Thus G(K/k] is isomorphic to the multiplicative group of Ck 3) We next exhibit a Picard-Vessiot extension where the differential Galois group is GLn(C). Let fc0 be any differential field and let K = kQ(Xι, . . . ,Xn). Let C = Ck0 = Cκ Suppose A = (αt j) is a non-singular n x n matrix over C. Then A Thus GL determines an automorphism of K by σ A (xf m ) ) = ΣaijXjm) n(C) is a subgroup of G(K/k$). Let fc be the fixed field of GLn(C). One sees that G(K/k) = GLn(C). Let

W(Xlt...,Xn)

'

We claim that L(Y) is a linear differential equation over k. To see this note that if A £ GLn(C) and X is the matrix such that W(Xι, ...,Xn) = \X\, then

while 1 0

0 Aτ

Thus L(Y) is invariant under σAl so L(Y) £ k{Y}. The elements X i , . . . , Xn are linearly independent solutions to I^^) = 0, thus K/k is PicardVessiot. In all of these examples the differential Galois group of the Picard-Vessiot extension is a linear algebraic group over the constant field. We next show that this is always the case. For the following arguments we fix K a very saturated differentially closed field. K will serve as a universal domain (ie. monster model) for all of our work. Let k be a differential field and let K/k be a Picard-Vessiot extension. Say K — k(uι,..., un) and L(Y) = 0 is the homogeneous linear equation determining the extension. Recall that since K/k is Picard-Vessiot, CK — Cfc We denote the common constant field C. Suppose k C F and σ : K -» F is an embedding fixing k. Then σ(u, ) is a solution of L(Y) = 0 for each i. Thus there are constants ct j £ CF such that σ(ui) = Σ,ci,juj- We call (cij) the matrix associated with σ. Theorem 8.1. There is Σ a system of equations in C[Xij.l < i, j < n] such that: i) If σ : K —* F is an embedding fixing fc, then the coefficients of the matrix associated with σ satisfy Σ. ii) If F D K and c £ CF satisfies Σ, then ut ι—*• ΣciJuj determines an embedding of K into F fixing k. This immediately yields: Corollary 8.2. If K/k is a Picard-Vessiot extension of order n, then G(K/k) is isomorphic to an algebraic subgroup ofGLn(C) (ie. G(K/k) is a linear algebraic group over the constant field).

proof of 8.1. Let p be the type of ¥ over k. Consider the map

φ : k{Yl9..., Y n } -> Kfaj determined by Yί ι—>• ΣJ=1 ZijUj

: I < ij < n]

and let Δ be the image of Ip under φ.

In other words, Δ is the ideal of polynomials in K[Z] such that if d is in the variety given by Δ and σ is the map tί, ι—> Σ d i j U j . Then
Let W be a vector space basis for K over C. For each / £ Δ, write /= where fw G C\Z}.

Let Σ be the ideal generated by the {fw : f G Δ, w G W}. Let L be the differential closure of K. Then CL is the algebraic closure of C. Since the elements of W are independent over C, and K and CL are linearly disjoint over C (as C is algebraically closed in K\ L ^ " for all constants c, if p(c) — 0, then for all w £ VF, p«/(c) = 0". By model completeness this is also true in K. Suppose σ : K —> K is an embedding fixing k and determined _by w, π-> ΣcijUj, for some constants c £ K. Then for every polynomial p(Z) £ Δ, p(c) = 0. By the above remarks, for all w £ W, Pu,(c) = 0. Thus all of the polynomials in Σ vanish at c. Let F D K and let A = (c, j) be a nonsingular matrix in CF such that c satisfies Σ. Let σ : K —>• F fix fc and send ιt, ι—»• ΣCJJ UJ. We claim that σ is an embedding. We chose Σ to insure that σ is a homomorphism. It suffices to show that σ is one to one. Suppose not. Then td(K/k) > td(k(σ(ΰ))/k) (if σ has a nontrivial kernel, then the Krull dimension of k(ΰ) is greater than the Krull dimension of k(σ(ΰ))). Thus td(k(ΰ,σ(u))/k(ΰ)) < td(t{ΰ,σ(π))/i{σ(π))). Also td(k(ΰyσ(u)))/k(ΰ) = td(k(ϋ,c)/k(ΰ)). But if constants c are algebraically dependent over a differential field L they are dependent over the constants of L. Thus td(k(u, c)/k(u)) = td(C(c)/C). But C is also the field of constants of k(σ(ΰ)). Thus

a contradiction. There is a beautiful Galois theory for Picard-Vessiot extensions. We state the main theorem here and refer the reader to the books by Kaplansky and Magid. Definition. Let K/k be differential fields and let G(K/k) be the differential Galois group. If H C G(K/k), let Fix(H) = {x G K : Vσ G H σ(x) = x}. We say that K/k is normal if for any x £ K\k there is σ € G(K/k) such that σ(x) ^έ x. Theorem 8.3. Let k be a differential field with Ck algebraically closed. If K/k is Picard-Vessiot, then K/k is normal, G(K/k) is a linear algebraic group over Ck and L H-» G(K/L) gives a one to one correspondence between the intermediate differential subfields of K/k and the algebraic subgroups of G(K/k). An algebraic subgroup H is normal if and only if Fix(H)/k is a normal. In this case

90

G(Fix(H)/k) is connected

is G(K/k)/H.

Moreover if k is algebraically closed, then G(K/k)

Much as ordinary Galois theory can be used to prove that the general quintic can not be solved by adjunction radicals, differential Galois theory can be used to prove the unsolvability of differential equations by simple means. Let f ( X ) G k{X}. We say that K is a Liouville extension of k if there are extensions k — KQ C K\ C C Kn = K, where each ίf, +ι is obtained from Ki by adjoining an integral, adjoining the exponential of an integral or making an algebraic extension. We say that f ( X ) = 0 is solvable by quadratures if it is solvable in a Liouvile extension. Theorem 8.4. Let k be a differential field of characteristic zero with Ck algebraically closed. Suppose that K/k is Liouville. If K D L D k is Liouville, then the connected component of G(L/k) is solvable. For example this method can be used to show that y' — y2 — x is not solvable by quadratures over C(x) References The algebraic Galois theory of Picard-Vessiot extensions is due to Kolchin ([Kolchin 4]). Kaplansky's Differential Algebra and Magid's Lectures on Differential Galois Theory provide extensive treatments of this subject. We refer the reader to these books for the proofs of theorems 8.3 and 8.4.

§9. Strongly Normal Extensions. In this section we will examine Kolchin's strongly normal extensions. This class of extensions contains the Picard-Vessiot extensions and also has an interesting Galois theory. Again we work inside a very saturated universal domain K. Definition. L/K is strongly normal if and only if i) CL = CK is algebraically closed ii) L/K is finitely generated iii) if σ : K —> K is an automorphism fixing K, then (L, CK) = (σ(L)> CK). For example, if CK is algebraically closed and L/K is Picard-Vessiot, we show that L/K is strongly normal. Suppose L = ^C(α), where a is a fundamental system of solutions to a linear equation over K. For any /^-automorphism

91

σ, <τ(α) £ (£,Ck), thus (i,Ck) 2 {σ(L),Ck) (σ(I/),Cκ) So equality holds.

Similarly, L is contained in

We will show that for strongly normal extensions G(L/K) is an algebraic group over Cκ Lemma 9.1. Suppose L/K is strongly normal and L = K(a). contained in the differential closure of K.

Then L is

Proof. Suppose not. Let F be the differential closure of L. Note that CF = CL — Cκ Let p be the type of α over the differential closure of K and let q be a non-forking extension of p to F. Since F contains no new constants, p is orthogonal to the the type of a new constant. Thus q is orthogonal to the type of a new constant. Let 6 realize q and let FI be the differential closure of F(b). Since q is orthogonal to the constants, CFI — CK> Since a and 6 realize the same type over K, there is an automorphism of K fixing K and sending a to 6. Thus since L is strongly normal, 6 £ (£,Cκ) In particular, there is a /^-definable function / such that c< = θ Λ / ( α , c ) = ϊ ) . By model completeness

Thus 6 £ {£, Cpi) = L. Thus α must be in the differential closure of K. Suppose L = K(a) and L/K is strongly normal. Since α is in the differential closure of K, there is a formula ψ(v) over K, which isolates the tp(a/K). Lemma 9.2. ψ(v) isolates tp(a/(K,Cκ)) Suppose 6 £ K, c £ CK and 0(ϊJ, 6,c) and -ι(t7, 6, c) split ψ(v). Then K |= 3c (/\ c = 0 Λ 3v3w (ψ(v) Λ V>(w) Λ φ(v,b,c) Λ -^(ΰf, 6, c))). By model completeness this is also true in the differential closure of K. But the differential closure of K has the same constants as K. Thus Ψ is not an atom over A", a contradiction. Before proving the general result we examine an important special case. Example. Weierstrass Equations: Fix 92, 93 £ Cκ with 27 g% - g\ φ 0. For α £ A let Gα(F) be the differential polynomial (Y')2 - α 2 (4Y 3 - (/2y - 93). We say that α £ K is Weierstrassian over A if it is non-constant and satisfies the equation Ga(a) = 0 for some α £ K. If K is the field of complex meromorphic functions , then the Weierstrass p-function p is Weierstrassian over K.

92

We assume that CK is algebraically closed. Consider the projective curve W given by the equation ZY2 = 4X3 - g2XZ2 - g3Z3. Since 270| - g\ / 0, W is non-singular and hence an elliptic curve defined over GK Hence there is an abelian group law on W. We write the group multiplicatively. W has a unique point at infinity (0, 1, 0) and this point is the zero of the group. In general (α,6, 1)""1 = (α,— 6, 1). [Note: Henceforth when we consider affine points of W we will use the standard affine coordinates.] ffG«(a) = 0, then (α, ^) G ^. We use the following lemma from [Kolchin 2]. Lemma 9.3. Suppose Gα(α) = 0 and G&(/?) = 0, where α and β are nonconstant. Suppose that (α, ^-)(/?, ^-) = (τ,<$) Then 7' = (α + 6)<5. In particular either 7 is constant or 7 is Weierstrassian over K with (7')2 = (α + b)2(463 — 926 — 93)Suppose a is Weierstrassian with Gα(α) = 0. Let L = K(a) and suppose that CL = C# Let σ : L —»• K be a /^-embedding. Consider Pσ = (σ(α), ^)(β, £)-' = (σ(α), ^)(α, -£). Pσ tW and Pσ = (0,1,0) if and only if σ is the identity. Suppose σ is nontrivial. Let Pσ = (01,02). By the previous lemma Cj = (α — α)c2. Thus GI is constant. It follows that 02 is also constant. Thus for every embedding σ there is Pσ G W(C K ) such that (σ(α), ^) = Pσ(α, ^). Thus σ(α) G (Z/,Cκ) so L/K is strongly normal. Suppose σ and r G G(L/K). Then P^ and Pr are in L. But the differential closure of L has the same constants as K, so these points are in W(Cκ} Then

Thus Pσr = PσPτ

Thus σ ι-* Pσ is an embedding from G(L/K) into

Let V7 isolate the type of a over K. The set {(ci, €2) G V7(C# ) : ^> holds of the first coordinate of (ci, C2)(α, ^f ))} is definable. Thus G(L/K) is isomorphic to a definable subgroup of W(Cκ) As W(Cκ) is an irreducible variety (and hence a connected group), the only proper definable subgroups of W(Cκ) are finite. Suppose G(L/K) is finite. Suppose β is in F the differential closure of L and 'φ(β) holds. Thus there is an automorphism σ of K such sending α to β. This automorphism corresponds to an action of the group W(Cκ) But then β is already in L. Thus in F there are only finitely many solutions of ψ, so a is algebraic over K. Thus we have shown that if CK = Cκ(a) and α is Weierstrassian over K it is either algebraic over K or G(/^{α), J^) is the group law of an elliptic curve over CK-

93

We will next show that the Galois group of a strongly normal extension is always an algebraic group over the constants. Let L/K be strongly normal and suppose L = K(a). Let ψ(v) isolate t(a/K). If ψ(b), then_ there is σ G G(K/K) such that σ(α) = 6. Since L/K is strongly normal, b £ (£,Gκ) In particular there is a /f-definable function g^ and c G GK such that gj(ά, c) = 6. By compactness and the usual coding tricks we can find a single /f-definable function g such that for all 6 G ψκ there is c G C*κ such that 6 = CjJ such that cEc0 if and only if /(c) = /(c0). Let G be the image of Y under /. Define on G by XQ - x\ = x^ if and only if there are CQ, ~c\ and ~CΊ G Y such that /(c, ) = z, and Λ*(co,cι,C2). Then (G, •) is isomorphic to G(L/K) and (G, •) is definable in the pure field structure of CF (Also CF = CL = Cκ^ In other words G(L/K) is isomorphic to a group definable in the pure algebraically closed field CK- The following theorem of van den Dries says that any such group is definably isomorphic to an algebraic group. Theorem 9.4. Let K be an algebraically closed field and let (G, •) be a group definable in K. Then G is definably isomorphic to an algebraic group over K. Thus we have proved the following theorem of Kolchin. Theorem 9.5. Suppose L/K is strongly normal and K is algebraically closed. Then G(L/K) is isomorphic to an algebraic group defined over CK-

94

Once we know that G(L/K) is an algebraic group over Cκ We can develop a Galois correspondence between algebraic subgroups and intermediate fields. Much of the Galois theory of theorem 8.3 generalizes For strongly normal extensions L/K, we will also study the group G((L,Gκ)/(lf, GK)) We will call this group the full differential Galois group and denote it Gal(L/K). The above arguments show that if L/K is strongly normal then Gal(L/K) is an algebraic group over GK In particular, there is an algebraic group G defined over CK such that G(L/K) = G(Cχ) and Gal(L/K) Ξ G(GK), (where for F D Cκ, G(F) denotes the F-rational points ofG). We will identify Gal(L/K) with G(Gκ) The above arguments show that there is a map 7 : Gal(L/K] —* G(Gκ) such that σ(α) G If (ά,7(σ)) and 7(σ) G K(a, σ(ά)> for all σ G Gal(L/K),. We next make a careful choice of the generator of L/K which will prove useful later. Definition. Let L/K be strongly normal and let F be the differential closure of K. We say that a G L is G-primitive if and only if α G G(Z/), L = If (α) and for all σ G G(F/K) α-lσ(α) G G(G*r). Lemma 9.6. Let K be algebraically closed. Every strongly normal extension L/K is of the form L = K(a), where a is G-primitive. Proof.

Since L is contained in the Differential closure of K and L/K is finitely generated, L/K has finite transcendence degree. Thus we can find a G L such that L = lf(α). For any σ G Gal(L/K), σ(a) G lf(ά,γ(σ))= lf(ά,7(σ)) and T(σ) G Ifjά, σ(α))= If (ά, σ(α)). Let 6, c realize t(a/K) such that 6,c are independent over L. Let r(α) j= 6. By the above remarks there is a rational function F over K such that F(Zz, 6) = 7(r). This F will work for independent realizations of the t(ά/K). In particular F(c,6) - F(α,c) = F(α,6). Let V be the if-variety such that α is the generic point of V. Then 6 is also a generic point of V over the field L(c). Thus the equation F(c,5F) F(α,c) = F(α,ϊc), must hold on a Zariski open subset of V. In particular we can find d G K such that F(c, d) F(α,c) = F((α, d) and ί^α, rf) G G(L) (Here we use the fact that if K is algebraically closed, L D K and V^ is a variety defined over K, then the If-rational point of V are Zariski dense in the L-rational points). We let a = F(ά,d). Let σ G Gal(L/K). We may as well assume that the c chosen above was independent of σ(ά) over If. Thus_ί(α,c/lf) = t(c,σ(a)/K). Hence F(σ(ά),d) F(c,σ(α)) = F(c,5). So JF(σ(ά),5) - F(c,σ(ά)) - F(α,c) = α. But F(c, σ(ά)) F(a,c) = γ(σ) and F(σ(α),rf) = σ(α). Thus σ(α) = α - γ(σ)" 1 , as desired. Finally we note that L = K(a). If α 0 If (α), there is r G Aut(K/K) such that r(αr) = α but r(α) / α. Thus σ = τ\L G Gal(L/K) and σ ^ 1. But since 1 σ(α) = α, α = α 7(σ)~ , so 7(σ) = 1 and σ is the identity.

95

The converse to 9.5 is also true. Lemma 9.7. Suppose K is a differential field with CK algebraically closed. Let G be an algebraic group defined over CK- Let F be the differential closure of K. Suppose there is a G G(F) such that for all σ G G(F/K) there is gσ G G(CK) such that σ(α) = α gσ. Let L = K(a^1). Then L/K is strongly normal. Proof.

l

l

l

Let σ G G(F/K), then σ(oΓ ) = gσ-ι a~ . Let ψ isolate t(a~ /K). Then

This sentence is still true in K. Thus for any automorphism σ, (£,Gκ) = (σ(L),Gκ) So L/K is strongly normal. Let Γ(K) be the coset space G(K)/G(Gκ) By elimination of imaginaries in K, we may assume that Γ(K) is a quantifier free G#-definable subset of K m . For any field L D Cκ let Γ(L) denote the L-rational points of Γ(K). Let p : G(K) —»• Γ(K) be the quotient map. If F is the differential closure of K, then T(F) = G(F)/G(CK) Lemma 9.8. Let a G G(F). Then α is G-primitive if and only if p(a) G T(K). Proof. Clearly p(ά) G Γ(tf) if and only if p(a) is fixed by all elements of G(K/K) if and only if α"1 σ(α) G G(CK) for all σ G G(K/K). By the last lemma this is if and only if a is G-primitive. Our next goal is to show that if G is a connected n-dimensional group, then T(K) is essentially Kn . This will require some background work. Let F/K be fields and let D(F/K) be the space of derivations of F which annihilate K. Let a?ι, . . . , xn be a transcendence base for F/K. Then ^ , . . . , -^ is a basis for D(F/K) as an F-vector space. First note that if αi, . . . ,α n G F and D = ΣXaf:> and D = 0, then for each i, D(x, ) = αt = 0. Thus ^|-, . . . , -£- are linearly independent. Next we consider the case F = K(xlj . . . ,x n ). If D G D(F/K) then for

K*iι - - - i *»)ι ^>(p(f )) - Σ £>(*.•)£- Th^ D = Σ^. )ror

In general if y is algebraic over K(x) with minimal polynomial p(x, y), then 0 = D(p(x,y)) = ΣD(xt)

+D ( y ) .

So

Thus there is a unique way to extend a derivation on K(x) to F. Thus is an n-dimensional F vector space.

D(F/K)

96 Let V C Km be an n- dimensional variety over K. Let K(V) denote the field of rational functions on V, K(V) = K\X}/I(V}. For p E V, let Op denote the local ring at p, ie. Op is the ring of rational functions defined at p. We choose affine coordinates at p so that #ι, . . . , xm E Op. We say that 6 : Op —» K, is a local derivation at p, if 6 is an additive homomorphism and <5(/ι/2) = f ι ( p ) δ ( f ι ) + /2(p) : Op -> Op we define a local derivation Dp by £>p(/) = D(f)(p). We let 7^(F) equal the set of all local derivations at p. Let /i, . . . , fn be generators for I(V). If δ is a local derivation at p, then Thus ό(#ι), . . . , <5(xm) are a solution to the system of equations

/3ft(p) - ίfc(p)\ (»ι, . ,ίΛn)

:

:

=0.

Thus ^(V) can be viewed as the tangent space at p. In particular if p is a simple point on V, then TP(V) is an n-dimensional vector space over K. Clearly each -^- : Op —+ Op. Let a ? ι , . . . , x n is a transcendence base for K(V)/K, then, by the above argument, the ^|- , . . . , -^- are linearly independent over K, and hence a basis for Tp(V). We next examine the case where G is a connected n-dimensional algebraic group. For α G G we let Tα : G —> G be the map x ι-» α#. For α,p E G, Tα induces Tα* : Oαp -^ Op, by Tα*/ = / o Tα*p. If D is a derivation of K(G)/K, then let TαDp be the local derivation at αp given by TaDp(f) = Dp(T^f). We say that D is invariant (actually left-invariant) if for all α,p G G, TαDp = Dαp. We let £(G) be the K-vectoτ space of invariant derivations. We call £(G) the Lie-algebra of G. Let 1 be the identity of G. We claim that £(G) is isomorphic to Tι(G) via the map D H+ £>!. We need only show that this map is surjective. Suppose δ E 7ί(G). We define D as follows. For / E K(G) we define /' by f'(x) = δ(T*f). Let D(f) = /'. We claim that D is a left invariant derivation.

So £) is a derivation.

97 Finally,

= Dax(f) since T^T* = TαV So D is left invariant. Thus £(G) is isomorphic to Tι(G), the tangent space of G at 1. Let F/K be of transcendence degree n. Say a?ι, . . . ,a?n is a transcendence base. We let ΩF/K be the F-vector space of differentials on F over K as introduced in §6. Then ΩF/K *s an rc-dimensional K vector space and dx\, . . . , dxn is a basis. In fact, ΩF/K ιs the dual space of D(F/K), ie. ΩF/K *s the space of F-linear maps from D(F/K) —> F. Each da? can be thought of as the map dx(D) = £>(*). If Φ : D(F/K) -> F, let K, = Φ(^). Let ω = EK^, then cj is induced by the map Φ. If V is a variety and p G V we consider the space of local differentials at p. This is the dual space of the tangent space Tp(V). Let ω be a differential of K(V)/K, say ω = Σgidfi. We say that ω is finite at p G V if all of the
1

i f f = i;

Suppose k C K and F is defined over k. We may choose the transcendence base x ι , . . . , x n such that x t G k(V). In this way we may assume that all of our bases are defined over k. If δ G D(K/k) and p G F, then 5 determines an element <5p of the tangent space of V at p by δj>(/) = δ(f(p)). If α; is a differential on V defined over k and well defined at p, then ωp(δp) is defined and in K. Then map δ »->• ωp(ίp) is a differential of K/k which we will call ω(p) the induced differential of ω at p. More specifically, if α; = Σ#*/« where G by τ(β)(x) = βxβ~l. In the manner we discussed above τ(β) induces automorphisms τ(βγ : K(G) -+ K(G) and τ(β) : C(G) -^ C(G). In general a map φ : £(Go) —» £(Gι) induces φ* mapping the invariant differentials on G\ to the invariant differentials on GO- Thus we have τ(β)* an automorphism of the invariant differentials on G. The next result shows the compatibility of the group operations with forming induced differentials from invariant differentials. We postpone the proof to Appendix B.

98

Theorem 9.9. Let a,β G G(K) and let ω be an invariant differential on G. Then ω(a - /?) = (τ(β)*ω)(a) + ω(β). In particular if G is abelian, then We now return to the following setting. K is an algebraically closed differential field and G is a connected algebraic group defined over CK- We let D be the derivation on K. CK plays the role of k in the above discussion. Lemma 9.10. Let α G G(K). Then α G G(Cκ) if and only if for every invariant differential ω on G, ω(a)(D) = 0. Proof. First, if α G G(Cjc), then for any / G CK(G) Π Oα, D(/(α)) = 0. Since, the space of invariant differentials has a basis of differentials defined over CW, this implies that every invariant differential vanishes at D. Conversely, if a 0 G(G#), then there is a local coordinate z, such that Xi(&) & GK The local differential dx{ on G translates to a local differential at 1 and this extends to an invariant differential ω on G. But then ω(α)(D) = D(xt(a)) φ 0. Corollary 9.11. Let α, /? G G(tf) Then α - β~l G G(Cχ) if and only if for every invariant differential ω on G, ω(a)(D) = ω(β)(D). Proof. Let 7 = α /J- 1 . By 9.8,for all ω

By 9.9, τ(β)*ω(y)(D) = 0 for all ω if and only if 7 G G(CK) (since r(^)* is an automorphism of the invariant differentials). If G is an algebraic group defined over CK. Let cji, . . . ,ωn be a basis for the invariant differentials on G, such that each ωi is defined over CK- For α G G let Λ t (α) = u>t (α)(£>). If α; = £#d/i where the g i y f i G Cκ(G\ then w(α)(D) = E^(α)ΰ(Λ(α)) τhus /< is definable in the differential field K. Let F : G(K] -* Kn by F(a) = (Λι(α)...Λ n (α)). By 9.10, F(α) = F(β) if and only if aβ~l G G(Cjf ). Thus the image of F can be identified with the quotient G(K)/G(CK) = T(K). In particular, Corollary 9.12 Γ(K) = G(K)/G(GK) can be embedded into K n . References All of the results in this section are due to Kolchin. They can be found in [Kolchin 2,5,6]. The proof of Theorem 9.5 that we give here is due to Poizat [Poizat 3]. Poizat 's book Groupes Stables contains Hrushovski's elegant model

99

theoretic proof of van den Dries theorem (9.4). The treatment we give here on G-primitives is taken from [Pillay-Sokolovic]. The basic results on derivations and differentials on algebraic groups can be found in [Rosenlicht 3]). The commutative case of Theorem 9.8 was proved by Rosenlicht while the general case is from [Kolchin 6].

§10.Superstable differential fields: We would like to prove the differential analogs of the following theorems about algebraically closed fields. We know that the theory of algebraically closed fields is quantifier eliminable and ω-stable. These results of Pillay and Sokolovic give partial converses. Theorem 10.1. i) (Macintyre-McKenna-van den Dries) If K is an infinite field and the theory of K admits quantifier elimination in the language of fields, then K is algebraically closed. ii) (Cherlin-Shelah) If K is an infinite field (possible with extra structure) and the theory of K is superstable, then K is algebraically closed. It would be natural to conjecture that any quantifier eliminable or superstable differential field is differentially closed. This question is open. We first note that the quantifier elimination question is subsumed by the sup erst ability question. Lemma 10.2. If T is a quantifier eliminable theory of differential fields (in the language of differential fields), then T is ω-stable. Proof. Let K \= T. By quantifier elimination any type over K is determined by the set of quantifier free formulas in the type. Thus an n-type is determined by the ideal of differential polynomials in K{X\,..., Xn} that vanish at a realization. Thus the number of types is equal to the number of prime differential ideals over K. By the Ritt basis theorem, every prime differential ideal is finitely generated. Thus there are only \K\ types over K. Thus K is ω-stable. In this section we will prove the following theorem from [Pillay-Sokolovic]. Theorem 10.3. If K is a superstable differential field with a non-trivial derivation (we allow the possibility of extra structure), then K has no proper strongly normal extensions. We begin by summarizing some of the Berline-Lascar [Berline-Lascar] theory of superstable groups which we will use in the proof.

100

Lemma 10.4. (Berline-Lascar) If K is a superstable field then for some ordinal a α and some natural number m, RU(K) = ω m. Definition. Suppose G be a superstable group and A C G is oo-definable. We say that A is α-indecomposable if A/H has only one class for any definable a subgroup H with RU(A/H) < ω . Theorem 10.5. (Berline-Lascar Indecomposability Theorem) If RU(G) = ωan and (Ai : i E /) is a family of oo-definable α-indecomposable sets each containing the identity of G, then the group H generated by the Ai is oo-definable and H is of the form A^1... Af*. Finally we recall Lascar's [/-rank inequality. Here 0 denotes the Cantor sum on the ordinals. Theorem 10.6. (Lascar's Rank Inequality): RU(a/Ab) + RU(b/A) < RU(a,b/A) < RU(a/Ab) φ RU(b/A). Let K be a saturated superstable differential field with RU(K) = ωam. By Theorem 10.1 ii), K is an algebraically closed field. The Cherlin-Shelah analysis of superstable fields also shows that any superstable field has a unique type of maximal rank. We call this the generic of K. Corollary 10.7. i) RU(CK) < ωa. ii) RU(x/x')<ωa. \\i) If A C K and a £ K is generic over A, then a' is generic over A. Proof. i) CK is an algebraically closed field, so K is an infinite dimensional vector space over Cκ. Thus for all n RU(K) > RU(C%) = RU(CK)n. Thus RU(CK) < ωa. ii) Clear from i) since CK is the kernel of the derivation, iii) By the (7-rank inequalities, RU(x/A) < RU(x, x'/A) < RU(x/Ax/) Since RU(x/A) = ωam, ii) implies that RU(x'/A)

φ RU(x'/A). = ωam.

Lemma 10.8. Let A C K and let α £ K be generic over A. differentially transcendental over A.

Then a is

Proof. Suppose not. Then we can find i and n such that αW is strongly algebraic over A,α(* + 1 ),... ,α( n ). Thus αW is algebraic over Aa(t+l). By 10.7 iii) αW is generic over A. Thus since α and αW realize the same type over A, α is algebraic over Aa''. But for any constant c G CK, RU(a H- c/A) φ RU(c/A) > RU(a + c,c/A) > RU(a/A). Since RU(c/A) < ωa, RU(a + c/A) = ωam, so

101

a -f c is generic over A. Thus t(a + c, a1 /A) = ΐ(α, a' /A). Since G/r is infinite this contradicts the fact that a is algebraic over Aa! . We can in fact prove something stronger. Lemma 10.9. Let A C K and lei a £ K be differentially algebraic over A, then RU(a/A)<ωa. Proof. We may without loss of generality assume that A |= Th(K) and (by taking forking extensions) that RU(a/A) = ωa. Let p = t(a/A). Let Φ0 = {x G K : x realizes p}. Fix 6 G Φo and let Φ = {x - b : x G Φ0}. Since p is stationary, Φ is α-indecomposable with respect to additive subgroups of K. For each x G K let Φ^ = xΦ. The Φx are α-indecomposable and contain 0. By 10.5 the additive subgroup H generated by the Φ^ is oo-definable and there are Xι , . . . , xn E K such that H = ΦXl + ΦX2 ... + ΦXn . Since xH C H for all x G K, H is an ideal. Thus H = K. Let y G K be generic over A(b, zi, . . . , xn)> There are yi, . . . , yn realizing p such that y = Σ,Xi(yi — b). But then, since the yt are differentially algebraic over A, y is differentially algebraic over A(b,Έ) contradicting the genericity oft/. We will prove that K has no proper strongly normal extensions. Let A be a very saturated differentially closed field containing K. It suffices to show that for G an algebraic group defined over CK if Γ(Λ) is the quotient space G(λ)/G(C\) and p : G(Λ) -* Γ(Λ) is the quotient map, then p maps G(K) onto Lemma 10.10. Let G be a connected n-dimensional algebraic group defined over GK Then RU(G(K)) = ωamn and every orbit of T(K) under 'the action of G(K) has (7-rank ωamn. Proof. The first remark is clear. More generally if V is an n-dimensional algebraic variety over a superstable field F, then RU(V) = RU(F)n. Let x G Γ(#). Let Stab(x) = {g G G(#) : 0z = x}. Let F be the differential closure of K. There is ft € G(F) such that h/G(CF) = «• Since CF = Cκ, 9 £ Stab(x) if and only if ft-^Λ G G(C^) if and only if g G hG(CK)h-1. Since Λt^(Cjr) < ωa, RU(Stab(x)) < ω<*. The orbit of x G Γ(Jf) under G(K) is isomorphic to G(K)/Stab(x). Using the ί7-rank inequality we see that each orbit has {/-rank ωamn. Since RU(G(K)) = ωamn and RU(G(CK)) < ωα By the ί7-rank inequality we must have β{7(Γ(/f)) = ωamn. Thus there are only finitely many orbits of T(K) under G(K). We will show that there is exactly one. In this case p : G(K) —> T(K) is onto and we are done. To prove this it suffices to show that Γ(^) has a unique generic type. By 9.12, Γ(Λ) C A n . Thus T(K) C Kn. But T(K) has rank ωamn = RU(Kn). Since Kn has a unique generic type, T(K) has a unique generic type.

102

References The material in this section is from [Pillay-Sokolovic]. Theorem 10.3 generalizes a theorem of [Michaux] who proved that a quantifier eliminable differential field has no proper Picard-Vessiot extensions. Poizat's book Groupes Stables contains treatments of the Berline-Lascar analysis of superstable groups and the Cherlin-Shelah results on superstable fields.

Appendix A: Seidenberg's Embedding Theorem In [Seidenberg 1,2] Seidenberg proved that any countable differential field can be embedded into a field of germs of meromorphic functions. This follows from an embedding lemma for finitely generated differential fields. Let Mer(U) denote the field of meromorphic functions on 17, for U C C open. Lemma A.I. Let K = Q(«ι -..un) and K\ = K(v). Suppose U is an open ball in C and τ : K —> Mer(U) is a differential field embedding. Then there is an open ball V C U and an extension of r to an differential embedding of K\ into Mer(V). Corollary A. 2. Let K be a countable differential field. Then K is isomorphic to a subfield of the field of germs of meromorphic functions at the origin. proof. By viewing K as a limit of finitely generated extensions and iterating A.I we can find a point x such that K can be embedded into the germs of meromorphic functions at x. By changing coordinates we may assume x = 0. The proof of lemma A.I, uses the following "primitive element theorem" from [Seidenberg 3]. Which we will prove shortly. Theorem A. 3. Suppose K is a differential field with a non-constant element. If u and v are differentially algebraic over K, then K(u} v) = K(u + λv) for some Proof of A.I. Let 0, = r(ui). By shrinking U we may assume that each 0, is analytic on U. Let a e U such that f(a) φ 0 for all / 6 Q(0ι, . . . ,gm) \ {0}. Changing coordinates we may assume that a = 0. Let gi(z) = ^ c« ,j^τ f°Γ z £ U (shrinking U if necessary). Note that
103 case 1. υ is differentially transcendental over K. Choose d0, d l 5 d 2 ,.. . £ C algebraically independent over Q(cί(J .i < nj G ω) and such that h(z) = Σ,dj^ converges on a neighborhood of V C U of 0. We claim that A, /ι', . . . are algebraically independent over Q{0ι . . .gn). Suppose p is a polynomial with coefficients in Q such that

Then P( c l,0> c l,l>

,
•- 5 C n , 0 , c n , l , • C n j / , d 0 )

i ^ m ) = 0.

Since the dj are algebraically independent p(c,Yb, .. ., Y m ) is identically zero. Thus by the isomorphism above

is identically zero. Thus h is differentially transcendental over Q(
Γ

r

1

is the minimal polynomial of t/ ) over KQ(υ,υ',υ( ~ )). Let d0, . . . , d r _ι be algebraically independent over Q(c,)J .i < n,j £ Since the Cfj are algebraically independent,

is irreducible. Let dr be a zero of it. Then |£ (c, do, , d r ) / 0. By the implicit function theorem there is W an open neighborhood of (c,do,...,d r _ι) and an analytic function F : W — > C such that F(c,d0, . . . ,d r _ι) = dr and p(w,f(w)) = 0 for all w G W. Consider the differential equation:

We can find a solution h which is analytic on a neighborhood of 0 such that for ΛW(0) = di for i = 0, . . . ,r. Sending v to Λ, gives r* an embedding of K\ extending τ\K0. Unfortunately, we might have τ*(u n ) = p* φ gn.

104

Let dj = h^\0) for j > r. By shrinking (and shifting) [/, we may assume that 0* is analytic on U and 0* = ΣC*^^-. Thus the map sending uf' to c, )<7 for i < n, w n (j) to c* j and v^) to c?; is a field isomorphism from KI to Q(CI,OI CM » - , c n _ι f o, CH-I,!, . . . , c*|0, c* f l , . . . , d0, di ,...)• Since Q(CI,OJ c l,l »

> C n _ι > 0 , Cn.1,1 , . . . , C* | 0 , C* f l , . . .) =

Q(CI,O,CI,I, -

j C n - i ^ j C n - i , ! , . . . , C n > o , C n | i , . . .),

we can find dj, d*, . such that Q(Cl,Oι CM> -

- , C n _ι > 0 , Cn-M >

> C n,0> cn,H

» ^0, ^1 , -

-) —

Let ΛI(Z) = X^dJ1^-. Let FI be a function analytic near (c, (/£, . . . ,c?r-ι) giving a branch of p — 0 such that F(c, d ) = d* . Then fti is the unique formal solution to y(r) = F(9l(z), g((z), g^(z), ..., gn_l(,), ffLi W,

, «n-ι(/) W, »,-.., ί^1-1^

with y(0) = cίj$> . . . , 2/Γ)(0) = d* . Since the initial value problem has a convergent solution near the origin, ΛI must converge on a neighborhood of 0. It is easy to see that mapping v to ΛI extends r to an embedding of KI into Mer(V) for some open ball V C C. We now examine the primitive element theorem. First, note that some assumption on K is necessary. If K contains only constant elements and L = K(u,v) where u and v are algebraically independent constants. Then clearly no it + Xv generates L/K. The proof uses the following lemma due to Ritt. Lemma A.4. Let K be a differential field and let £ G K with £' ± 0. Let G(X) € K{X} be nontrivial of order n, then there are rational numbers c0, . . . , cn such that ' Proof. Suppose not. Let H(X) be of minimal order r such that for all rationale c 0 , . . . , c n fΓ(Σ>ί'') = 0. Let Λ ( Y 0 , . . . , Y Γ ) e ϋf[F] such that H(X) = ' ' n+1

n

Since y vanishes on Q , g is identically zero (as Q is Zariski-dense in K ). Thus -jfc = 0 for each j. Thus for j = 0, . . . , r n

dh dUj ~_

105 For j = 0 we get

du

^ 0 = 0

and for j > 0

From this we see that the vectors

are linearly dependent. By lemma 4.1 they are linearly dependent over Ck Thus r Σ biξ = 0 for some constants 6 0 > > &r where not all of the 6t are zero. Since ξ is algebraic over CK (by 2.1), ξ £ CR , a contradiction. Proof of A.3. Consider K(u,v)(X). be irreducible such that

u + vX is differentially algebraic over K(X). Let G

G(X, X',..., X< Γ >, (« + vX), ...,(« + »*)<•>) = 0

(1)

and s is minimal. Let 10 = ti + vX. For i < * fjjg = 0. While for i = β fffly = v. Implicitly differentiating (1) with respect to X^ we get

SG

dG

Because of the minimality of G, ^f^y is not identically zero. By lemma A.4, we can find λ G K such that 55^7(λ, ti + vλ) ^ 0. Using (2) we see that

Appendix B: The proof of 9.8 In this section we will give Kolchin's proof of Theorem 9.8. First suppose G and H are algebraic groups defined over an algebraically closed field K, f : G —*• H is rational and x G G. As usual we have /* : K(H) -> K(G) by /*0 = gof. This in-turn induces / : TX(G) -+ Tf(x)(H), by fδ(9) = *(/*y) Using the isomorphisms between the tangent spaces and the Lie-algebras we obtain f f : £(G) -* £(ff). In particular if ί E TX(G) and let D G £(G) be such that ^ = ί, then /*£) is the element E of £(#), such that Ef(x) = /δ.

106

Lemma B.I: Suppose / : G —> H is a homomorphism, then ff depend on x.

does not

proof: Let δ G Tχ(G) and let D G C(G) be such that Dx = δ. Let D = f$D. For A G 0ι.

Suppose E = f£~D. Then

Thus Dι=EιsoD = E. If / : G-> G! and g : G, -> G2, then (0 o /)£ - 9*(x)<>ft Lemma B.2: a) If / : G —> # is constantly c, then // = 0. b) Let Tv be left multiplication by v, then (Tυ)f is the identity on £(G). proof: a) Let δ e TX(G) and let D G £(G) be such that Dx = δ. Let E = f$D. Since c G # , Ej(x)(h) = f δ ( h ) - δ(Λ(c)) = 0 for all A G O/φ. Thus £7/(ar) is the trivial tangent vector. So E is the trivial derivation. b) This is clear since (Tv)fDx

= Dvx for D G £(G).

We fix v a point on G. To simplify notation we will refer to the maps /# as /. [If for a particular map (a non-homomorphism) it is important which tangent space we use to define the map we assume we use the base point υ.] We consider the following maps: -ti, j'2 : G -^ G x G by ή(ar) = (a:, v), i 2 (a?) = (v, x). -Δ : G —» G x G is the diagonal map x ι-» (x, x). -π1? π2 : G x G —* G are the projections, 7Γi(a?ι, x 2 ) = χi -i : G —»• G is the identity. -e : G —> G is the zero map. -A,; : G —> G is right multiplication x \-> xv. 1 -ψ : G x G -> G by τ^(x, y) = xy" .

107 l

-for υ G G τ(v) : G —> G by conjugation, x ι—> v~ xv. Lemma B.3: C(G x G) = ή£(G) Θ *2£(G). proof: Clearly ij is injective. Suppose iiD -\- i2E — Q. Then

+ TΓ!^, by B.2 a) iιD + i2£) = 0. Similarly £ = 0. Thus iι£(G) Θ 22£(G) has twice the dimension of C(G), and hence is equal to£(GxG). Lemma B.4: Δ = i\ + 2*2proof: Let D G £(G). We will show that for all g G #(G), (ΔD-iιD-i2D)^g = 0. Since AΓ(G x G) = πjίf (G) 0 πj/f (G), this implies (ΔD - i^D - i2D) = 0. First, suppose / G Oi, then

=π

=

(iD -iD - cD)

= 0. Now let g e Os. (ΔD - iiD - i 2 D)πίy(s,ί) = (ΔD - UD - i

= (ΔD - tiD - i = 0 (by the claim above). Thus for all g G K(G), (ΔD - iiD- i2D)πιg = 0. The same is true for π 2 . Thus by the above remarks, for all D ΔD - iiD - i2D = 0. So Δ = ιΊ + t 2 . Lemma B.5: Xv o ^ = ττι — π 2 . proof: For any x £ G λυ o ψ o i\(x) — x. Thus Xv o ψ o iι is the identity map 2. On the other hand for any x, λvoψo Δ(x) = v. Since this map is constant, the it induces the trivial endomorphism of the Lie-algebra. By lemma B.4, \v o ψ o 2*2 = λ υ o ψ o (Δ — 2*1). By the above remarks, this is —i. Thus (λυ o ψ + π2 - τrι) o ii = i + 0 - i = 0 and (λv o ψ + τr2 - πi) o f 2 = -l + t - 0 .

108

Thus by lemma B.3, λv o ψ = πi — π z. Lemma B.6: λ v = τ(v). proof: Xv = Tυτ(v), but Tv acts on £(G) as the identity. For δ G T>(K/k), we define a tangent vector at υ lδ(v), the logarithmic derivative by lδ(v)(g) = δ(g(v)). If / is any rational map, then, of course Lemma B.7: lδ(xv) = r(υ)lδ(x) + lδ(v). proof:

= λ υ oψlδ(xv,v)

Finally, suppose ω is an invariant differential on G. If x G G, α;.,. is the local component of u; at x. We defined the induced differential ω(x) on Ί)(K/k], by In particular if x, υ G G(lf ), α;(a?t;)(ί)α;art,(/6(xt;)) By B.7 this is w.-i.vWt;)/^)) +«,(/«(«)), which is τ(υγωf(lδ(x))+ω,(lδ(υ))

= (τ(»)'ω(*) + ω(w))ί.

Thus we have proved: Theorem 9.8: If x,v G G(K) and ω is an invariant differential on G, then ω(xv) = r(v)*t4;(ar) +ω(t;).

Appendix C: Kolchin's Irreducibility Theorem

This appendix is devoted to the following theorem of Kolchin. Theorem C.I. Let K be an algebraically closed field with derivation D. Suppose V C Kl is an irreducible algebraic variety defined over K. Then V is £)-irreducible.

109

Suppose V is an irreducible variety. Suppose (x,y) £ V is a generic point of V where we (without loss of generality) we may assume that xι , . . . , xn are algebraically independent and y\ , . . . , ym is algebraic over K(x}. For a = 1, . . . , m let pi(Έ, Y) be the minimal polynomial oίyt over K(x). An easy induction shows that for all n, (j)Pi_(Έ

θy

\ _

"~

J /- ^

'''

-(j)

/

./J- 1 )^

' ' '** '' *

for some polynomial rt j with coefficients in K. If (af, y) is a D-generic point of V (ie. a point of maximal Morley rank in K), then #, a?,^2), ... are algebraically independent and K(x,y) = K(x)(y). Thus there is a unique D- generic type. Since an irreducible algebraic variety has a unique D-generic type there is a unique D-irreducible component of maximal rank. We will need to do a bit more work to show that there is only one Z>-irreducible component. Suppose L D K are differential fields and α, b £ Ln . We say that a i—> 6 is a differential specialization over K if /(6) = 0, whenever / £ K{X} and f(ά) = 0. We will use the following lemma on specializations. Lemma C.2. Let K be an algebraically closed field with derivation D. Let V C Kn be an irreducible variety defined over K, p £ /(V), and let α £ V be a /^-rational point. There is a differential field extension L D K and β an L-rational point of V such that p(β) ^ 0 and there is β H-> α is a differential specialization over K. proof: If dim V > 2, let H be a hyperplane through α not contained in V(p). Let W be an irreducible component of V Π # through a. Then dim V7 = dim V —I and p ^ I(W). Thus without loss of generality we may assume V is a curve. If V is not smooth there is a smooth curve W and a polynomial map σ : 1 W —>• V. Let α* £ W Π σ~ (α). Suppose there is a differential field L D K and /?* an L-rational point of W such that p(σ(β)) / 0 and /?* K-> α* is a differential specialization over /ί. Let /? = σ(/?*). If (7 is any D-closed set defined over /f and β G AT, then β* £ σ-1^. Hence α* £ σ-1^ and α £ C. Thus /? ι-> α is the desired specialization. Thus without loss of generality we may assume V is a smooth curve. Let Oa be the local ring of regular functions at α and let Ma be the maximal ideal of functions vanishing at a. Since V is smooth Mα/M2 is a one dimensional K-vector space. Let t £ Mα be a generator for Mα/M2. Let K(V) be the function field of V, there is a unique derivation £) : K(V) —»• if (V) extending the derivation on /f with D(£) = 0.

no There is a natural embedding of K(V) into the field of formal Laurent series K((t)) sending Oα into K[[t]] and Ma into tK[\t]]. Consider the derivation δ defined on if ((*)) defined by

i—m

Clearly δ(K[[t]]) C K[[t]] and 6(tK[[t]]) C tK[[t]]. Since there is a unique derivation from K(V) to K((t)) extending D and sending t to 0, we must have D : Oa -> Oa and D : Mα -» Mα. Let 7Γ : Oα -» if be the evaluation map / »-> /(α). If / G Oα, then for some α G if, and g G Mα, f = a + g. Then

since D(g) G M. Since D(α) = D(πf), π commutes with D, thus π is a differential specialization. Let L = K(V). Let β = (a?ι,.. .,a? n ) G ^ be the coordinate functions. Clearly π(β) = α. Since p is not identically 0 on 7, /?(/?) φ 0. We now give the proof of C.I. Let pi and rt-}J- be the polynomials described above. If (af, y) is any point of F, then p, (ϊί, Vj) = 0 and _(- \ _

J

-(j)

/- -i

/ ίy ' ^'^'''''' ^*

for all j. Let t=l

For any /(X, Y) G iί{X,Y} there is a polynomial y with coefficients in K and natural numbers s and < such that if p(x,y) / 0, then

Suppose (ϋ,ϊ7) G V is D-generic and f(ΰ^) = 0. Then g(ΰ,ϋ! ', . . . , tϊ(β), y) = 0. Since ϊίjϊί', . . . ,tl^5) are algebraically independent, (/(TXX^Y) = ft0(X, Y)hι(X, Y) where Λ 0 G K\X, . . . ,^,7] and_Λι_G ^[X,F] and /n G K\X,Ϋ] vanishes on all of V. It follows that if / G K{X, Y] vanishes at the D-generic of V, then / vanishes on {(x,y) G V : p ( x , y ) / 0}.

Ill

Since for any a £ V, we can find L C K and an //-rational point β in V \ V(p) with β h-» a a differential specialization, it follows that any f(X,Ϋ) which vanishes on the Z)-generic of V vanishes on all of V. Thus if W is the D-irreducible component containing the £)-generic, we must have V = W.

References Books J. Baldwin, Fundamentals of Stability Theory, Springer Verlag, 1988. M. Hirsh and S. Smale, Differential Algebra, Academic Press, 1974. A. Buiiim, Differential

Algebra and Diophantine Geometry, Hermann, 1994.

I. Kaplansky, Differential E. Kolchin, Differential

Equations, Dynamical Systems and Linear

Algebra, Hermann, 1957. Algerbra and Algebraic Groups, Academic Press, 1973.

S. Lang, Algebra, Addison Wesley, 1971. S. Lang, Introduction to Algebraic Geometry, Interscience, 1959 . I). Lascar, Stability in Model Theory, Longman Scientific &; Technical, 1987. A. Magid, Lectures on Differential ciety, 1994.

Galois Theory, American Mathematical So-

B. Poizat, Cours de Theone des Modeles, Nur al-Mantiq wal-Ma'rifah, 1987. B. Poizat, Groupes Stables, Nur al-Mantiq wal-Ma'rifah, 1985. J. Ritt, Differential

Algebra, Dover, 1950.

(i. Sacks, Saturated Model Theory, Benjamin, 1972. Papers [Berline-Lascar] C, Berline and D. Lascar, "Superstable groups", Annals pure and applied logic, vol 30, 1986. [Blurn] "Differentially closed fields: a model theoretic tour", in Contributions to Algebra, ed. H. Bass, P. Cassidy and J. Kovacic, Academic Press 1977. [Brestovski] M. Brestovski, " Algebraic independence of solutions of differential equations of the second order", Pacific J. Math, vol 140, 1989. [Cherlin-Shelah] G. Cherlin and S. Shelah, "Superstable fields and groups", Annals Math. Logic vol 18, 1980. [(ίramain 1] F. Gramain, "Non-minimalite de la cloture differentielle I: la preuve de S. Shelah", Theories Stables, Paris Seminar Notes 1980-83.

112 [Gramain 2] F. Gramain, "Non-minimalite de la cloture differentielle II: lapreuve de M. Rosenlicht", Theories Stables, Paris Seminar Notes 1980-83. [Hrushovski-Sokolovic] E. Hrushovski and Z. Sokolovic, "Minimal subsets of differentially closed fields", Trans. AMS, to appear. [Kolchin 1] E. Kolchin, "Existence theorems connected with the Picard-Vessiot theory of homogeneous linear ordinary differential equations", Bulletin AMS vol.54, 1948. [Koichin 2] E. Kolchin, "Galois Theory of Differential Fields", Amer. J. Math vol 75, 1953. [Kolchin 3] E. Kolchin, "Constrained extensions of differential fields", Advances in Math, vol 12, 1974 [Kolchin 4] E. Kolchin, " Algebraic matrix groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations" Annals of Math vol 49, 1948. [Kolchin 5] E. Kolchin, "On the Galois Theory of Differential Fields", Amer. J. Math vol 77,1955. [Kolchin 6] E. Kolchin, "Abelian extensions of differential fields, Amer. J. Math, vol 82, 1960. [Lascar 1] D. Lascar, "Les corps differentiellement clos denombrables", Theories Stables, Paris Seminar Notes 1980-83. [Lascar 2] D. Lascar, "Relation entre le rang U et le poids", Fundamenta Math. vol 121 (1984). [Macintrye-McKenna-van den Dries] A. Macintyre, K. McKenna and L. van den Dries, "Elimination of quantifiers in algebraic structures", Advances in Math vol 47, 1983. [Marker] D. Marker, "Introduction to the model theory of fields", this volume. [Michaux] C. Michaux," Sur Γelimination des quantificateurs dans les anneaux differentiels, Comptes Rendus 47, 1983. [Pillay] A. Pillay, "Differential algebraic groups and the number of countable differentially closed fields", this volume. [Pillay-Sokolovic], A. Pillay and Z. Sokolovic, "Superstable differential fields", JSL 57, 1992. [Poizat 1] B. Poizat, "Deux remarques a propos de la propriete de recouvrement fini", Journal of Symbolic Logic, vol 49, 1984. [Poizat 2] B. Poizat, "Rangs des types dan les corps differentiels", Theories Stables, Paris Seminar Notes 1977/78. [Poizat 3] B. Poizat, "Une theorie de Galois imaginaire", Journal of Symbolic Logic vol 48, 1983.

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[Poizat 4] B. Poizat, "C'est beau et chaud", Theories Stables, Paris Seminar Notes 1980-83. [Rosenlicht 1] M. Rosenlicht, "The non-minimality of differential closure", Pacific J. Math, vol 52, 1974. [Rosenlicht 2] M. Rosenlicht, "On Lioulville's theory of elementary functions", Pacific J. Math, vol 65, 1976. [Rosenlicht 3] M. Rosenlicht, "A note on derivations and differentials on algebraic groups", Portugal Math vol 16, 1957. [Seidenberg 1] A. Seidenberg, "Abstract differential algebra and the analytic case", Proc. AMS vol. 9, 1958. [Seidenberg 2] A. Seidenberg, "Abstract differential algebra and the analytic case IP, Proc. AMS vol. 23, 1969. [Seidenberg 3] A. Seidenberg, "Some basic theorems in differential algebra (characteristic p, arbitrary), Trans. AMS vol. 73, 1952. [Shelah] S. Shelah, "Differentially closed fields", Israel J. Mathematics vol 16, 1973. [Wood] C. Wood, "The model theory of differential fields revisited", Israel Journal of Mathematics, vol 25, 1976.

Differential algebraic groups and the number of countable differentially closed fields Anand Pillay University of Notre Dame

Introduction. We give an exposition of several results on definable groups in differentially closed fields, and applications thereof. Among other things we give a proof of the result [HS] that there are continuum many countable differentially closed fields of characteristic 0. The theory DCFQ (of differentially closed fields of characteristic 0) is complete and ω-stable and thus by [SHM] has either < KQ or 2N° countable models. But until recently it was not known which. Rather surprisingly it turns out that classical mathematical objects, specifically elliptic curves, lie behind the existence of continuum many countable models (or at least behind the present proof). One of the essential points is to find some strongly regular nonisolated type which is orthogonal to the empty set. The required type p is found inside a suitable definable (in DCFo) subgroup G (of finite Morley rank) of an elliptic curve E(ά) with differentially transcendental ^-invariant α. So it turns out that there are "exotic" groups of finite Morley rank definable in differentially closed fields. In any case in section 2 of this paper we prove the existence of 2N° countable differentially closed fields. The argument we present was sketched for us by E. Hrushovski, although we have a few additional simplifications. In fact, given an example due to Manin [M], showing that for any elliptic curve E there is differential rational homomorphism from E onto Ga (the additive group), the existence of the required type p turns out to rather a direct matter, requiring neither the deep Zariski-geometry interpretation, nor the properties of "jet groups" of algebraic groups. On the other hand, in so far as simple (noncommutative) groups of finite Morley rank are concerned, no exotic structures are to be found in differentially closed fields. Any such group G will be definably isomorphic to an algebraic group living in the constants. This is exactly the finite Morley rank case of Gassidy's Theorem [C2], of which I will give an easy proof in section 1. This implies that any infinite field F of finite Morley rank definable in a differentially closed field K is definably isomorphic to the field of constants of K. The remaining part (namely the infinite Morley rank case) of Cassidy's Theorem, states that a simple group of infinite Morley rank definable in a differentially closed field K is definably isomorphic to an algebraic group over K. We were unable to find

115 a "naive" proof of this (as in the finite Morley rank case), but in section 5 we outline a fast proof due to Buium. In sections 3 and 4 we present the machinery and properties of "jet" groups of algebraic groups. This is due to Buium [Bl] and [B2], who worked scheme-theoretically. Following Buium, we recover Manin's results [M] concerning the existence of differential rational homomorphisms from arbitrary abelian varieties into vector groups. Following [HS] we point out how this, together with the "Zariski-geometry" interpretation, yields the classification, up to nonorthogonality, of nontrivial Morley rank 1 types in DCF0. For the remainder of this paper U = (U, +, , δ) will denote a big saturated differentially closed field of characteristic 0. C denotes the field of constants of U. All objects we talk about will be ones definable in U. RU will denote Lascar's "[/-rank". Morley rank ( RM ), RU etc., will always mean in the sense of (U, +, , <$), unless stated otherwise. We usually write x1 in place of δ(x) The reader is referred to Marker's paper [Mr] in this volume for various basic facts about differentially closed fields. But I remark here that the field of constants C with all the definable structure induced from U is simply an algebraically closed field (G, +, ). For the purposes of this paper a differential algebraic group is simply a group definable in a differentially closed field. (The equivalence of the categories of differential algebraic groups and definable groups is pointed out in [PI]. However to obtain an equivalence which preserves "fields of definition" is rather more tricky, and appears in [P2].) By a minimal group we usually mean a definable commutative group without proper definable connected subgroups (where "definable" means in U or in the field structure of U, depending on the context). I will be using facts from stability theory and stable group theory quite freely. (See [Po].) Among other things, I may be using facts such as : an infinitely definable group in an ω-stable structure is definable; an infinite definable group in an ω-stable structure has an infinite definable commutative subgroup. I also use fairly freely the fact that any group definable in an algebraically closed field (A', +, •) is definably isomorphic to G(K) for G some algebraic group defined over K, and also that an infinite field definable in (AT, +, •) is definably isomorphic to K. We also make use in various places of the following theorems about abstract algebraic groups and abelian varieties (see for example [Sh] and [L]): (a) If G is a connected algebraic group defined over a field fc, then G has a unique maximal normal linear algebraic subgroup N, and G/N is an abelian variety. (b) if A is an abelian variety then the torsion part of A is infinite, and for any n, the n-torsion of A is finite. (c) if A is an abelian variety defined over k, then any algebraic subgroup (connected or not) of A is defined over acl(fc). At some points (as in (a), (b), (c) above for example) we will be interested in objects defined in U just in the field language. We will thus denote the structure (ί/,+, ) by U-., and we talk about "definable in U~". Similarly tp~(a/k) is the complete type of α over k in the structure U~, and RM~(α/fc) is the Morley

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rank of tp~ (a/k) in the structure U". For notions from stability (including the "geometric" theory), I would recommend [P3]. Simply for basic stability I recommend [Las]. I thank E. Hrushovski for several communications, and also D. Marker for some helpful discussions.

§1 Simple groups of finite Morley-rank. The key point in getting a handle on simple definable groups (of finite Morley dimension or not) is to embed them definably in some GL(n,U). In fact any group definable in U can be embedded in an algebraic group over U, namely into a a group definable in U~ (this is proved in [P2]). For groups of finite Morley rank the proof is a little easier and we give it now. Lemma 1.1. Let G be a connected group of finite Morley rank definable in U. Then G is definably embeddable into some definable group H where H is definable in U~~ and connected (as a group definable in U~).

Proof. Let k be a countable model (elementary substructure of U, or equivalently differentially closed differential subfield of U) over which G is defined. Let a be a generic point of G over k. As RM(α/fc) is finite so is tr.degree k(a)/k. Thus there is some tuple α from k(a) such that k(a) = k(a). Write a = /(α) where f ( x ) is some ^-definable function defined at α. Choose b £ G generic over k(a). So k(b) = £(/(&)). Then α 6 is generic in G over each of k(a),k(b). Moreover α ft € (k(a))(b) = (*(/(«))(/(*))• So /(α.6) G *(/(α),/(6)). Letp(x) = tp~(a/k) (= tp-(f(b)/k) = tp-(f(a.b)/k)). Then easily /(α) and /(&) are independent realisations of p in U". Similarly for /(α) and /(α 6), /(6) and /(α 6). Let g(x, y) be a function fc-definable in U^ such that /(α.6) = g ( f ( a ) , f ( b ) ) . Clearly then g is (in U~) a generically associative fc-definable function from p x p to p. By a result of Hrushovski [Po, 5.23] (or even WeiPs theorem) there is some connected group H definable over k in U~ such that p(x) is the generic type of H. The map / which takes generic α of G to /(α) £ H can be easily seen to extend to a definable embedding of G into H. Corollary 1.2. Let G be a centreless connected group of finite Morley rank definable in U. Then there is a definable embedding of G into GL(n,U) for some n.

Proof. Let H be a group definable in U~ such that G is definably embedded in H. H can be identified (definably) with an algebraic group over U. H may be assumed to be connected. Choose H of least dimension (or equivalently least Morley rank in U"). Then Z(H), the centre of H, has trivial intersection with G,

117 so G embeds definably in H/Z(H) (another connected group definable in U~). Thus Z(H) is finite (otherwise άim(H/Z(H)) < dim(#)). But then H/Z(H) is centreless. Thus we have definably embedded G into a centreless algebraic group which we call H again. But it is well known [B] that any centreless algebraic group embeds (as an algebraic group, so definably in U~) in GL(n,U) (some n). This completes the proof. We need to know some elementary facts about definable subgroups of U n and (U*) .

n

n

Fact 1.3. Let G be a definable (in U) subgroup of U . Then G is a vector space over G. If moreover G has finite Morley rank, then G is a finite-dimensional vector space over G. Proof. The set A = {a £ G : αG C G} is a definable (in U) additive subgroup of G which is infinite (as it contains Z). Thus (as G is strongly minimal in U), A = C. The last remark is clear, for if the G-dimension of G is > m then RM(G) > m (since G-dimension (G) = RM(G) if the latter is finite). Fact 1.4. The map which sends (zi, . . . , xn) to (^ , ...., morphism from (U*)n onto Un with kernel (G*)n. *

-) defines a homo-

Proof. The fact that the map is a homomorphism with kernel as stated is checked immediately. Surjectivity follows for example by comparing RU-ranks (as U is differentially closed, so unstable). Theorem 1.5. Let G be a simple group of finite Morley rank definable in U. Then there is a group H definable in the structure (G, +, •) such that G is (in U) definably isomorphic to H. Otherwise stated; there is an algebraic group H defined over G such that G is definably isomorphic to H(C). Proof. By Corollary 1.2, we may assume G is a definable subgroup of GL(n,U) for some n. Now G, as an ω-stable group, has an infinite commutative definable subgroup A. We use some elementary facts on linear algebraic groups, for which the reader is referred to [Bo]. By the Lie-Kolchin Theorem, we may assume that A is a group of upper triangular matrices. Let p be the homomorphism from A into the group D of diagonal n x n matrices (namely p is simply projection on the diagonal), p is clearly definable. Case (i). Ker(p) is nontrivial. Let B = Ker(p). B is then a commutative group of unipotent matrices, and is known to be isomorphic by the map

118 to a subgroup, say #1, of the additive group of n x n matrices over U. Then BI is definable and of finite Morley rank. By Fact 1.3 Bl is a finite dimensional vector space over C. In particular B\ and thus also B, are connected. As G is simple and of finite Morley rank, by Zilber's indecomposability theorem, there are 0(1), ..,g(k) in G such that G - B - Bg(l) -Bg(k). Thus G is definably m isomorphic to a group H C C /E (where E is some definable equivalence relam tion). But any definable (in U) relation on C is definable in(G, +, •). Thus G can be identified with a group definable in ((7,4-, •), as required. Case (ii). Ker(p) is trivial. n Thus p yields an isomorphism of A with a subgroup DI of (U*) . If D\ Π n (C*) is infinite, then an application of Zilber's indecomposability theorem as in Case (i) again yields the desired conclusion. Otherwise let m be the cardinality n n of DI Π (G*) . Then clearly mDl Π (G*) = 1. Let D2 = mDi. So by Fact 1.4, n D2 is definably isomorphic to a (infinite) subgroup of U , which must have finite Morley rank and is again a finite-dimensional vector space over C. Proceed as in Case (i). This completes the proof. Corollary 1.6. If F is an infinite field definable in U and F has finite Morley rank, then F is definably isomorphic to the field C of constants of U. Proof. It is known that F must be algebraically closed. PSL^F) is then a simple group of finite Morley rank definable in U, so by 1.5, is definably isomorphic to a group H definable in C. Now F is definably isomorphic to a field definable in the pure group structure of PSL2(^) (by considering a Borel subgroup). Thus F is definably isomorphic to a field K living in C. Then K is definable in (G, +, ) so is (by [Po]) definably isomorphic to the field C. The result follows. Remarks 1.7. (i) We were a little heavy handed in the proof of 1.5. From Facts 1.3, 1.4, simplicity of G and the facts used about commutative linear groups, one sees directly that G is nonorthogonal to G, thus internal to C. (ii) Cassidy points out in [Cl] that if G is any connected definable subgroup of (C/*)n and GI is the Zariski closure of G then GI Π (G*)n = GΠ (G*)n. One can deduce from this that if such a group G has RU-rank ωm for some m > 0, then G is algebraic. (iii) Proving that any definable simple group G of infinite Morley rank is definably isomorphic to an algebraic group over U, is a rather more subtle issue. One can assume that G is Zariski dense in a simple algebraic group. But the kind of arguments used in the finite Morley rank case do not work, as U is not a "pure field". Cassidy's proof in [C2] involves detailed facts about root systems in Che valley groups. Buium [B2] has a direct and conceptual proof using the "jet groups" which appear in the next sections, together with the fact that a simple algebraic group acts irreducibly on its Lie algebra. We sketch his proof in section 6. It would be nevertheless nice to find a more model-theoretic proof, for example by finding a "large" definable diagonalisable subgroup of G, and then

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using (ii) and the indecomposability theorem. On the other hand it is not true that any simple differential algebraic subgroup of GL(n,U) of infinite Morley rank is already algebraic (namely definable in U~). For example let G = {(X, Xi) : X G SL(n, U)} with multiplication (X, X') (y, Y') = (ΛΎ, (AY)')- G can be represented as a definable subgroup of Gi(2n, U), f with X in the top left hand corner and bottom right hand corner, and X in the top right hand corner.

§2. Elliptic curves and many countable models 2. Elliptic curves and many countable models. The aim here is to give as painless a proof as possible of the existence of continuum many countable differentially closed fields. We will give such a proof, modulo a result of Manin (Lemma 2.3 below). In fact Manin's result is essentially just an example in the introduction to [M]. There are several definitions of "elliptic curve". For example : a connected one-dimensional algebraic group which is complete as an algebraic variety, a nonsingular projective algebraic curve of genus one with a distinguished point, or even a nonsingular projective cubic curve with a distinguished point. Among the important things for us is that the family of such objects has an infinite moduli space. In any case an elliptic curve is a certain kind of algebraic group, and as such is an object defined in U~. (We view U~ as a universal domain for algebraic geometry. If you wish identify U~ with C, the complex field.) More generally an abelian variety is a connected (infinite) algebraic group whose underlying variety is complete (see [Sh]). An abelian variety is said to be simple if it has no proper abelian subvarieties. An elliptic curve is then just a one-dimensional abelian variety. The following fundamental information can be found in any basic text on elliptic curves, e.g [Si]. Fact 2.1. To each elliptic curve E can be associated an element j(E) G t7, with the following properties. (i) j(E) is in any field of definition of E (in the algebraic sense). (ii) E is isomorphic to E\ iff j(E) = j(E\) (Here isomorphic means as algebraic groups). (iii) for any j, there is an elliptic curve E such that j(E) = j and E is defined over j. Example 2.2. Let α,6 G U satisfy 4α3 + 2762 φ 0. Then the solutions of the equation y2 = x3 + ax + b together with the point at infinity form an elliptic curve (whose 0 is the point at infinity, and with the "chord-tangent" group law). The j-invariant of this curve is traditionally given as (123)(4α3/4α3 + 2762).

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Any elliptic curve is isomorphic to one of the above form. Given j φ 0,123, the following cubic defines an elliptic curve with invariant j: y2 = 4z 3 -27(j/j123)z-2707.;-123). The following appears essentially in the introduction to [M]. Lemma 2.3. Let E be an elliptic curve. Then there is a definable (in U) nontrivial homomorphism from E into (U, +). Let us quickly remark that if X is a set definable in U", then the Morley rank (or U-rank) of X in U, is simply ωd, where d = RM~(X). In particular, an elliptic curve has RU-rank u;, as an object definable in U. Corollary 2.4. Let E be an elliptic curve defined (in U~) over k. Then there is a subgroup G of E, definable in U such that (i) G has finite Morley rank, (ii) G is infinite, connected and fc-definable, and has no proper infinite kdefinable subgroup. Proof. Let / : E —> U be the homomorphism given by lemma 2.3. Now in U, E has U-rank ω (and is still connected). Thus ker(f) is a proper definable subgroup of E, hence has finite RU-rank (equivalently finite Morley rank). As (U,+) is torsion-free it follows that ker(f) contains Tor(£") (the torsion part of E). Note that Toi(E) is infinite. Let B be the intersection of all definable subgroups of E which contain Tor(.E'). Then B is fc-definable, infinite, connected, and of finite Morley rank. Now choose G to be a fc-definable infinite connected subgroup of B which has no proper infinite fc-definable infinite subgroup. The main point is to show that for suitable elliptic curves £?, any group G as given by 2.4 has generic type orthogonal to 0. Theorem 2.5. Suppose α 6 U is differentially transcendental. Let E(a) be an elliptic curve defined over α (in U") and with j(E(a)) = a. Let G be a subgroup of E(a) of finite Morley rank, which is connected, infinite, defined over α (in U), and has no proper infinite definable subgroup also defined over α. Let p(x) be the generic type of G. Then p(x) is orthogonal to 0. Proof. Let us write p(x) as p(x,a). Basic facts about orthogonality mean that we have to prove: Restatement: if 6 6 U, tp(b/d) = *p(α/0) and 6 is independent from α over 0, then p(x, α) is orthogonal to p(ar, 6). Aiming for a contradiction, we assume this fails. Thus we have 6 as in the hypothesis of the restatement, but with p(x,α) nonorthogonal to p(x, 6).

121 Step I. We find a connected group H defined over some parameter c with c independent from a over 0, and a definable isomorphism h between H and G. Note p(x, a) is the generic type of G. By assumption p(x, a) is nonorthogonal to p(x,b). Let G(6) denote the "copy" of G over 6. Let {6 0 ,6ι,...} be a set of realisations oftp(a/d) such that {α,&o,&ι,...} is 0-independent. Claim la. There is a "small set" A of parameters, there is a proper α-definable subgroup N of G, and there is n < ω such that G/N Cdcl(AUG(6 0 )U.. .UG(6n)). Proof. This is really a basic stable-group-theoretic result due essentially to Hrushovski. However, I do not know a reference for the specific form in which the claim is stated, so I sketch the proof. The nonorthogonality assumption means that there is a model M containing {α,6}, and there are elements c realising p(x, ά)\M and d realising p(x, 6)|M such that c forks with d over M. Let e be the canonical base of tp(dy M/c, a). Then e G dcl(c, α)Π dcl(d, M, di, MI, ..,dn, Mn) (for some n) where (d0,M0) = (d, M) and (dt ,M, ) is an {α,c}-independent sequence of realisations oftp(d, M/c, α). Let 6, G M, be the copy of 6. Then clearly {α, 6 0 , . . . , bn} is an independent set of realisations oftp(a/d) so we may assume that the 6, are the same as the ones mentioned before the claim. Also note that c (as well as e) is independent from MQ U ... U Mn over α. Let G, = G(6, ). Note that dt G G(6, ). Write e = f ( c ) for some α-definable function /. Let N be {g G G : for ci realising p(x,a)\{a,g}, f(g c x ) = f(cι)}. Then (as e G acl(α)) one can show that N is a proper α-definable subgroup of G. Let X be a big Morley sequence inp(ar,α)|(Mo U ... U Mn). Then one can show that for ci realising p(z,α)|(M0 U ... U Mn U X), cι/N G dcl({α} U X U {/(ci c2) : --j^n}- Thus c is independent from α over 0. Now by choice of G and the fact that N is α-definable, N must be finite. At this point we could replace G by G/N and E by E/N. Alternatively, let hi

122 be the map from H into G defined by hι(y) = Σh~l(y). Using the fact that both G and H are divisible, with finite n-torsion for all n, it is clear that ΛI is a surjective definable homomorphism with finite kernel N±. Now N\ must be acl(c)-definable (by the torsion condition on H). Thus hi induces a definable isomorphism between H/N\ and G, where H/N\ is definable over c\ G acl(c). By the claim ci is independent with α over 0. Thus Step I is complete. Step II. From H, c and h as given by Step I, we construct a commutative group Hn defined over Q(c) in U~, and a surjective homomorphism Λ I , definable in Urn, from Hn tp E. Let r G S(Q(α)) be the generic type of G, and r~ the reduct of r to U~. So r~ is precisely the generic type of E (over α). Note that RM H is finite. By Lemma 1.1 (and its proof), we may assume that H is a subgroup of a group defined over fco = Q(c) in U~. In particular the group operation on H is definable in U~ over fco Let k be some differential field containing α,c such that the isomorphism Λ is defined over k (in U). Let b be a generic point of H over k. Then Λ(6) is a generic point of G over k. Moreover, by quantifier elimination, there is some n, such that Λ(6) G Ar(6,&',.., b^) (the field generated by k together with {6,6', ..,&(n)}). Thus h(b) = ΛI(&,&', ..,b^) where h(y) is Ar-rational. Let q~ = tp~(6,δ', ..,&( n )/Q(c)). As in the proof of Lemma 1.1, mulitiplication on H induces a fco-rational generically associative map * from q~ x q— to q—, yielding as there a (commutative) group Hn say, definable over fco in U~, and with generic type <j~, and with group operation agreeing with * generically. It is clear that for 6 realising g~~|fc, ΛI(&) realises r~\k. It should also be clear that for generic fc-independent (in the sense of U~) realisations 6, c of q~ |fc, ΛI(& c) = Λι(6) ΛI(C). Thus ΛI defines a fc-rational isomorphism from Hn onto E. Step III. We find a quotient group B of Hn, definable over acl(Q{c)) in U~, which is definably (in U~) isomorphic to G. This part of the proof just involves facts about algebraic groups. We now use the fact that Hn (obtained in Step II) has the structure of a (commutative) algebraic group defined over kQ = Q(c), and that ΛI is a rational homomorphism from Hn onto the elliptic curve E. Now Hn has a unique maximal connected linear algebraic subgroup L, and Hn/L is an abelian variety. As there is no nonzero rational homomorphism from a linear algebraic group into an abelian variety, it follows that L < fcer(Λι). Note that (by uniqueness) L is defined over fc0 (in U~), thus Hn/L is also defined over Λ0 Now if A is an abelian variety defined over fco then any algebraic subgroup of A is defined over acl(fc0) Thus in particular, we see that ker(h\) is defined over acl(fco). Let B = Hn/ker(hι). an Thus B is an algebraic group defined over acl(fco) d ΛI induces a rational isomomorphism h^ say between B and E. Thus B is also an elliptic curve, whereby j(B) = j(E) (by 2.1 (ii)). By 2.1 j(E) G Q(α)Π acl(Q(c)). But Q{α) is independent from acl(Q{c)) over 0 in U, so the same is true in U~, thus α = j(E) G acl(Q), which is a contradiction to the choice of α. This contradiction proves Theorem 2.5.

123 Corollary 2.6. DCFQ has 2*° countable models. Proof. Let E(a) be an elliptic curve defined over α (in U~) and with j-invariant α, where α G U is differentially transcendental over 0. Let G — G(a) be a subgroup of E(a) given by Corollary 2.4. Let p(x) G S(a) be the generic type of G. By Theorem 2.5, p(x) is orthogonal to 0. We will show that DCF0 has "ENI-DOP". Formally T having "ENI-DOP means that there are models M 0 ,Mι,M 2 ,M such that MQ C MI, M0 C MI, MI is independent from M2 over MO, M is prime over Ml U M2, and there is a strongly regular stationary "eventually nonisolated" type pi G S(M) such that p is orthogonal to MI and orthogonal to M2. (A stationary type q is said to be eventually nonisolated if there is a finite set A and a stationarisation of q over A which is nonisolated.). To obtain this, let MQ be some model independent with α over 0. Let 6,c be independent generics of U over MQ such that 6 + c = α. Then {α,6,c} is pairwise independent over 0, and also pairwise independent over MQ. Let MI be prime over MO U {6}, and M2 prime over MO U {c}. In particular α is independent from each of MI, M2 over 0 . Thus p(x) is orthogonal to each of MI, M2. Let M be prime over MI U M2. Then α G M. We can find types pi, ..,pm over M each of Morley rank 1 such that p(x) is domination equivalent to pi ® ®pm (this uses the fact that p(x) is the generic type of a group of finite Morley rank.) Thus each pi is orthogonal to each of MI, M2. Now G is divisible. So as a structure in its own right (namely with all α-definable structure induced from U) G is not No-categorical. It easily follows that some pi is eventually nonisolated. (In fact the fact that G has no proper infinite connected α-definable subgroups implies that G is almost rank 1, namely that after adding finitely many parameters, there is a Morley rank 1 (not necessarily degree 1) subset X of G such that G G άcl(X). As G is not N0-categorical, some strongly minimal subset of X is not No-categorical, yielding the required ENI-type.) So pt witnesses ENI-DOP. By [SHM], DCFv has continuum many countable models. Remark 2.7. As we shall see below, if E is an elliptic curve, then any infinite definable (in U) subgroup of E contains Tor(#), and thus E has a unique minimal infinite definable subgroup (of finite Morley rank). This is also true if E is a simple abelian variety, namely an abelian variety without proper connected nontrivial algebraic subgroups. Thus the group G in 2.4 can be taken to be minimal (namely without proper connected definable subgroups). In fact making use of the Zariski interpretation, one can show (as we do later) that G must be actually strongly minimal. In the remainder of this paper we develop tighter connections between groups definable in U and those definable in U~. In particular we obtain (following Buium [B2]) a generalisation of Lemma 2.3 to arbitrary abelian varieties. This is connected with Manin's "Theorem of the kernel" from [M].

124

§3 Jet Groups. In this section we develop the theory and some properties of the "twisted jet groups" of Buium [B2], but working in the definable category. We work as before in the big differentially closed field U. As motivation let us first consider the general linear group G = GL(n,U). If the matrix X = ( x i j ) G G, let X1 denote the matrix whose ijth coordinate is x'ij. The set {(X, X') : X G G} has a natural group structure: (X, X') (Y, Yf) = (XY, (XY)'). It is rather clear that this group is precisely the subgroup of GL(n,U) consisting of matrices

(X

(o

X'\

x)

where X € G. The Zaxiski closure of this group in GL(2n,U) is the set of matrices

(X

Y\

(o

x)

with X G G and Y arbitrary. Let us call this group GI. We have a natural projection map p : GI —> G whose kernel is the group of matrices

where Y is an arbitrary n x n matrix. Let us call this kernel LI. Note that LI is isomorphic to the vector space Un . In fact LI is precisely
(x o \ (o x) with LI. Let us call the first group G^i (a copy of G). The action of GI,I on LI by conjugation in GI is exactly the action of G on (//(n,U) by conjugation, inducing the "adjoint" representation. We also have a map / : G —> LI defined in DCFQ, obtained by taking X to (X, X1) G GI and then projecting onto LI . (/ will not be a homomorphism, but a crossed homomorphism. ) It can be checked that / is precisely the map X *-> X~1X'. Thus Ker(f) is G(G) = GL(n,G), the group of G-rational points of G. By comparing with [Bo. AG 16 and 1.3], we see that GI is precisely the tangent bundle of G. Note that the process can be iterated, to obtain linear algebraic groups G2,Ga,... where for example G2 is the group consisting of matrices

with X G G, y, Z arbitrary. These groups are the natural jet space groups attached to G (higher versions of the tangent bundle).

125

We proceed to develop these objects in greater generality. As mentioned above, we work in the definable category (namely we do not concern ourselves with the geometric structure of objects). The constructions below generalise arguments in the proof of 2.5. Definition 3.1. Let G be a connected group definable in U, defined over k C U. So a point of G is some n-tuple from U. If α = (α l 5 .., α n ) G Gthen by a' we mean the tuple (α^, . . . , α{,). (i) for m > 0, em(G) denotes the group ({(α,α', ..,α(m)) : α G G},*(m)) where the group operation *(m) is defined by :

(So eo(G) is precisely G). We also let em denote the map from G to em(G) Remark 3.2. (i) For any ra > 0, em(G) is also a connected group definable in U over fc, and em is a fc-definable group isomorphism. (ii) Suppose that the group operation on G is definable over k in U~ , namely there is a partial function /( , ) defined over k in the field language, such that for α,6 G G, /(α,6) = α 6. Then for any m, the group operation *(m) is also defined over k in U~ . For example this is the case if G itself is definable in U~ , or if G is a subgroup of such a group. As mentioned earlier RM"~ denotes Morley rank computed in the structure U~ . For the remainder of this section we assume that G is a group definable over k in U~, which is connected in U~, and thus also connected in U. (The latter fact is not obvious. It is due to Kolchin (see appendix C of Marker's paper [Mr] in this volume. The special case we use here is also proved in [HS].) Construction 3.3. We construct, for each m, a group Gm also definable over k in U~, and surjective homomorphisms πm : Gm —> G m _ι. Forra = 0,Gm = G. Let m > 1. Let α be a generic point of em(G) over k (generic in the sense of U). (Note that a is then of the form (0,0', . . . ,α(m)) = em(α), where α is some generic point of G over fc, in the sense of U.) Let pm(x) = tp(a/k)ί and p^(x) — tp~(a/k). So p~ is uniquely determined and is moreover a stationary type. Let us write simply . for multiplication in em(G). Let α,6,c be an independent (in U) set of realisations of pm. Then (α 6) c — a (b - c), and (α 6) is independent from c over k etc. Now working in U~ , α, 6, c are also independent realisations of p^ over fc, and the above independence facts remain true. Also by 3.2 (ii), the group operation is definable over k in U~ . So a result of Hrushovski [Po, 5.23] (or equivalently WeiΓs theorem) yields a connected group Gm definable over k in U~ , whose generic type is p" , and whose group operation agrees with . on independent realisations of p~ . So we have defined the groups Gm. To obtain the homomorphisms πm, note that, for all α G G and in particular for generic

126 such α, e m _ι(α) is a subtuple of e m (α). Define τr m (e m (α)) = e m _ι(α). Then clearly p m _ι is a map, defined over k in U~ , from the realisations of p^ onto the realisations of p^-n which is "generically" a group homomorphism from Gm to G m _ι. Namely for independent realisations α, 6 of p^, τrm(α 6) = πm(α) ττ m (δ). Thus πm extends to a surjective homomorphism from Gm to G m _ι, defined over

The above construction yields an identification of the generic points of βm(G) with certain generic (in U~) points of Gm (via the identity map). We would like however to identify in a U~-definable manner all of e m (G) with a subgroup of Gm. This is actually quite straigtforward, and can be obtained (as we show now) through the Hrushovski construction of Gm as the group of "germs" of U~-definable generically defined maps from p^ to pm— generated by the maps fa : b —> α*6 for generic independent realisations α, 6 of p^ (Here we let * denote the group operation on em(G).) In particular G "is" the group of such germs of the form /βl /βa . Let now O be the set of {(α, 6) : α, 6 both realise p~ , a * 6 is defined, and for generic c realising p~ , (α * 6) * (c) = α * (6 * c)}. Then Concludes {(α,6) : α,6 are generic points of em(G)}. Define an equivalence relation ~ on O, by (α, 6) ~ (αi , 61) if for generic c realising p~ , α * (6 * c) = a\ * (61 * c) (if and only if (α, 6) and (αi , 6χ) define the same germ, namely the same element of Gm). So we see that Of ~ "is" a subset of G m . Let X = {c : for some (α, 6) G O, α * b is defined and equals c, and for generic c? realising p~, (c * cf) * d~"1 is defined and equals c}. Then clearly em(G) G -X" and X is oo-definable over k in U~. Define a map / from X into G m , by: if c — α * 6 for (α, 6) G O, then /(c) = (α, &)/ ~. Clearly / is well-defined (for if c = αi * 61 for (αι,6ι) G O then for generic d realising p~ , (α * 6) * d = c * d — (αl * 61) * rf). On the other hand / is 1-1, for if /(c) = /(ci), then for generic d realising p~ , we must have c* d = ci * d. By the second clause in the definition of X, we conclude (after "multiplying" by d~l) that c = GI . Now / is fc-definable in U~. By compactness we can find sets XQ D X and ^b C G m , both definable over fe in U~, and a bijection g between XQ and Yb, also defined over Ar in U~ , such that the restriction of g to Xis precisely f . So the restriction of the map g to em(G) defines a group embedding into Gm extending the identity map on p~ . By means of this embedding we can and will assume that em(G) is actually a subgroup of G m , and thus that em is an embedding of G into G m . If the reader is unhappy with this, he or she can work with elements of 9(em(G)) (g as above) in place of em(G). (Note that it is trivial to find an embedding of em(G) in Gm definable in U, but the question here was to find one "definable in U-".) Lemma 3.4. (i) RM-(Gm) = (m + l)RM-(G), for all m. (ii) If X is a subset of G which is definable in U, then for some m there is a subset Y of Gm definable in U~ such that em(X) = Y C em(G).

127 (iii) If H is a (connected) subgroup of G which is definable in U, then for some ra there is a (connected) subgroup Hm of Gm, definable in U~ and such that em(H) Proof. (i) Suppose RM~(G) = n. Let a be a generic point of G over k in the sense of U. Then clearly n = differential transcendence degree of k(a) over k. In fact n of the coordinates of a are differentially transcendental over k, and the rest are algebraic over these together with k. So clearly the transcendence degree / m of *(α,α ,...,α( ))/t) = (m + l)n. Thus RM"(ip-(e m (a)/t) = (m + l)n. As tp~(em(ά)/k) is a generic type of Gm it follows that RM~(G m ) = (m + l)n. (ii). By quantifier elimination in JDGFo, there is some m < ω and some formula ψ(xQ,xl,...,xm) (with parameters) in the language of fields such that m X = {α G G : U j= V>(α,α', ...,α( )}. So simply let Y be the subset of Gm defined by ψ(x). (ίii). This does not appear to follow directly from (ii). We may assume H is defined over k. Let again m < ω and φ(xoJ..>xm) be a formula (over k) in the language of fields such that φ(x1xlj ...,χ(m)) defines jff in U. Then again em(#) = {(«o,*ι,-,*m) G Gm : (a?o, ,*m) € e m (G) and φ(xQ,..,xm)}. Let 6 be a generic point of H over k. Then e m (fr) is a generic point of em(H) over fc. Let g~(x) = tp~(em(b)/k). As in Construction 3.3, the realisations of q~ in Gm are closed under generic (in U~) multiplication. Thus basic stable group theory yields a connected subgroup Hm of G m , definable in U~ over k, and with generic type q~. Claim. Hm Πe m (G) = em(H). Proof of claim. Clearly em(H) C Hm (as any element of em(H) is a product of generics of e m (ίf), and all generics are in Hm. For the other inclusion it is enough to show that if c = (c, c',..., c(m)) is a generic point of Hm Π em(G) over k (in the sense of U) then c G em(H). Let c be such. It is then easy to see that tr.degree(c/fc) > tr.degree(e m (6)/fc) (where remember em(6) was a generic point of em(H) over k). On the other hand, as c G Hm and q~ is (the unique) generic (in U~) type of #m, it follows that RM~(c/fc) = tr.degree(c/fc) = tr.degree(^ ) (=RM(g")). Thus tp"(c/t) = q~. In particular U |= ψ(c), namely U |= ψ(c, c',..., c(m)). So c G em(H).

128

§4 Vector groups, the Buium-Manin homomorphism and rank 1 types. We begin to make more use of notions from algebraic geometry. The reader is referred to [Sh]. Recall the notation from section 3 : given a connected group definable in U"~ , we have groups GI, GI . . . definable in U~ and rational homomorphisms ττm : Gm x Gm-ι> Let τm be the induced homomorphism from Gm onto G = G0. Lemma 4.1. Suppose Gis an algebraic group defined over fc, of dimension n (namely RM~~(G) = n. Then for all r, ker(τr) is a vector group of dimension r.n. Proof. We just sketch the proof. We first do it for the case r = 1. Let V be an affine open neighbourhood of the identity in G, V C U m , some m, such that the identity element e of G is the origin (0, ..., 0). Assume G is defined over k. Let Oe denote the local ring of G at e, and m e its maximal ideal. We also assume that the first n coordinates of x = (a?ι, ..,xm) G V form local coordinates (or parameters) for G at e. Namely the coordinate functions a?ι, ...,xn form a basis of the fc-vector space me/ml. One also knows that xι,..,z n generate me (in Oe). Thus there are αt j G k such that x E V has the form

Namely for j = n + 1, .., m,

As any element of ml is of the form ΣfiXi for /, 6 m e , this means that for j = n + 1, ..., m, ay = £ α. ja?i + Σ ΛJ *» (where f i j is in me). (*) We now want to bring in the group GI . Let x denote a point of Um . With a little work we can consider V\ = Zariski closure in U2m of {(#,#') : x G V}, as an affine open subset of GI, with (0,0) the identity of Gl. Let O C V x V = {(x>y) £ V x V : x y € V}, where x - y refers to multiplication in G. Then the group operation on O is given by a rational function /(-,-) (defined over k). The function which takes ((α, α x ), (6, 6')) to (α - fr), (α - 6)') for (α, b) G O, is also a rational function fι say defined over k. Then we can assume that whenever (α,αι) G Vί, (6,61) G Vϊ and the product (in GI), (α,αι) (6,61) G Vi, then this product is /ι((α,αι), (6,61)). Now α?ι,..,x n are also differentially independent parameters over k, so generically {xi, ..,xnιx(, ...,xj,} is algebraically independent over k. Thus we can choose an affine open neighbourhood of the identity i n G i , apoint of which has the form (xi,..,x n ,x n + 1 ,..,a? m ,ti,..,< n ,/ n + 1 ,..,< m ),

129

where the xβ satisfy (*), a?ι,..,a? n »*ι> »>*n are local parameters at the identity (0, ..... ,0) of GI, and for j = n + 1, .., m, tj is of the form:

»=ι

»=ι

t=ι

*'

(**)

where /, j = /tf j(a?ι, --.j^n) € τπe, and (/, j is some rational function of (arι, ...,ar n ,^ι, ...,tn). Now such points (z,t) with x = (0, ..,0), clearly form an open neighbourhood of the identity in the algebraic group Arer(τrι) C GI. By (**) any such point has the form: Qff|mt<).

i=l

(***)

1=1

On the other hand, it is well-known (see [L]) that (working back in the original affine-open neighbourhood V of G) for (generic) x,y £ V, and i = 1, ..,n, (x y), = (xi + ι/, ) mod M2, where M is the maximal ideal of the local ring at the identity of G x G. It follows from this together with (* * *) that for generic (0,t),(0,β) in fcer(τrι), (0,<) (0,β) = (0,* + *). Thus generically Jfeer(τrι) is isomorphic to the group U n . So fcer(ττι) is isomorphic to U n , as required. Suppose the lemma is proved for r. The kernel of the projection (Gr)ι —»• Gr is a vector group, by what we have just shown. On the other hand this clearly factors through pr+ι : GΓ+ι —>• G Γ . So ker(πr) is a vector group (of dimension = dim(G)). Composing with τr completes the proof. The dimension assertions are contained in 3.4 (i). Lemma 4.2. Let A be a connected commutative algebraic group. Let B be a connected definable (in U) Zariski-dense subgroup of A. Then A/B is definably isomorphic to a subgroup of Un ( for some n). In particular B contains the torsion part of A. Proof. Let AQ — -4,-Aι,... be as in section 3. By Lemma 3.4 there is some m such that em(B) = Bm Π em(A). As B is Zariski-dense in A, clearly τm\Bmis onto A. Let Lm = ktr(τm). Then Am = BmLm. As (by 4.1) Lm is a vector group, Bm Γ\Lm has a complement L in L m and Am is the direct product of Bm and L. Let π be the corresponding projection map from Am onto L. Let / be the homomorphism A —> L defined by: /(α) = π(α,α', ...,α(m)). m m Then α 6 *er(f) iff (α, α',.., α( >) G Sm iff (α, α',., α( )) e em(B) iff a G J3. Note that this lemma recovers Remark 1.7 (ii). Corollary 4.3. Let A be a simple abelian variety. Then A has a unique minimal infinite connected definable subgroup. Proof. As A has no proper connected nontrivial algebraic subgroups, every infinite definable subgroup of A is Zariski-dense. In particular, by 4.2, any infinite connected definable subgroup of A contains the torsion part oΐ A (which is

130 known to be infinite). Thus the intersection of all connected definable subgroups of A is infinite, and is definable (by ω-st ability). Nothing we have said so far shows that proper definable subgroups exist. We proceed to show that one can always find definable subgroups of finite Morley rank (of abelian varieties). As in Buium's treatment we require the following result of Rosenlicht [R. Lemma 3]: Fact 4.4. Let A be an abelian variety, B a vector group, and G an extension of A by B. (Namely we have an exact sequence of algebraic groups:

Then there is a connected algebraic subgroup G\ of G such GI projects onto A and dim(Gι) < 2dim(A). Proposition 4.5. Let A be an abelian variety. Then A contains an infinite definable subgroup B of finite Morley rank. Proof. Let Am be as in section 3. Let τm be the projection Am —> A (composition of the TΓj). For each m there is a unique minimal algebraic subgroup Bm such that Bm projects onto A under ττm . (Uniqueness is by the fact that if B\ , B^ both project onto A then each of A m /£?ι, Λ m /S 2 embeds in the vector group Lm = fcer(rm), whereby Am/(Bι Π #2) embeds in a vector group, so B\ Π B^ projects onto A. The last implication is due to the fact that there is no nontrivial homomorphism from a vector group into an abelian variety.) By uniqueness we conclude that ττm maps Bm onto £ m _ι. On the other hand Fact 4.4 says that dim(£m) is bounded by 2dim(A). Let D = {a £ A : (α, α', ...,α< m )) G Bm for all m}. Then D is a definable subgroup of A. Claim. D has finite Morley rank. Proof. Assume everything is defined over k C U. The bound on the dimensions of the 5mmeans that tr.deg(fc(α, α', ...α(m), ...)/&) is finite, for any α (Ξ D, and thus RM(tp(a/k)) is finite for any α G D. Thus RM(D) is finite. Putting together 4.3 and 4.5 we have: Corollary 4.6. Let A be a simple abelian variety. Then A contains a unique smallest definable nontrivial connected subgroup of finite Morley rank. Lemma 4.7. Suppose A, B are simple abelian varieties, and G, H are definable subgroups of A, B respectively. If G is definably isogenous to H, then A is rationally isogenous to B (namely isogenous by a map definable in U~).

131 Proof. Without loss of generality, H is definably isomorphic to G via the isomorphism / : H —>• G. Assume everything is defined over k. We use the notation of section 3. Let 6 be a generic point of H over k. Then for some n and for some fc-rational function /i, f ( b ) = /ι(6, 6', —,6^). b is a generic point of B over fc (in the sense of U~) and /(6) is a generic point of A over fc (in the sense of n U~) (as G is Zariski-dense in A and H is Zariski-dense in B). Also (δ, &', .., &( )) is a generic point of the group Hn over k (in the se nse of U~). Thus clearly /i gives rise to a surjective fc-rational homomorphism from Hn onto A. Let τn denote the canonical Ar-rational surjection Bn —» j£. Then rn|#n : Hn —» # is surjective. Let L — ker(τn\Hn). So L C ker(τn) and the latter is, by 4.1 a vector group. As there is no nontrivial rational homomorphism from a vector group into an abelian variety, it follows that L C ker(fι). Thus f\ induces a rational map from Hn/L onto A. As Hn/L is rationally isomorphic to 5, we obtain a rational homomorphism h from B onto A. As B is a simple abelian variety, h is an isogeny (namely has finite kernel). This completes the proof of 4.7. We now bring in some rather heavier model-theoretic facts. Fact 4.8. [HS, HZ]. Let X be a strongly minimal set definable in U. If X is not locally modular then a strongly minimal field is definable in Let pc denote the generic type of the constants Cu of U. pc has Morley rank 1. Corollary 4.9. Let q be a non locally modular type of RU-rank 1 (in U). Then q is nonorthogonal to pc> Proof. Let X be a strongly minimal set in q. By 4.8 some infinite field (of finite Morley rank) F, is definable in Xeq. By Corollary 1.6, F is definably isomorphic to Cu. This clearly suffices. Finally we describe a relationship between nontrivial types of Morley rank 1 which are orthogonal to pc and simple abelian varieties which are not rationally isomorphic to algebraic groups defined over Cu . We are interested in such types up to nonorthogonality, and some groups up to rational isogeny. First let q be such a type. By 4.8, q is locally modular and hence by [H], there is a strongly minimal group G whose generic type is nonorthogonal to g, so is without loss q itself. By 1.1, let A be an algebraic group in which G is definably embedded. As G is commutative and strongly minimal, we may assume that A is commutative, connected with no proper connected algebraic subgroups. A is then either a linear group, or a simple abelian variety. If A is linear, then the proof of Theorem 1.5 shows that G is definably isomorphic to a group living in Cu, contradicting the nonorthogonality of q to pc- Thus A is a

132 simple abelian variety. Note that G must be the unique minimal finite Morley rank definable subgroup of A. On the other hand, let A be a simple abelian variety not rationally isomorphic to an algebraic group defined over GU Let G be the definable subgroup of A given by 4.6. Then G is connected, infinite, and has no proper infinite definable subgroups. Thus G is almost strongly minimal. Let X be a strongly minimal subset of G. If X is not locally modular, then by 4.8 the generic type of X is nonorthogonal to pc. In particular the generic type of G is nonorthogonal to pc- As in the proof of 2.5 (Step I), G is definably isomorphic to a group H living in C^j. As in Step II of the proof of 2.5, A is rationally isomorphic to an algebraic group defined over GU, contradicting our hypotheses on A. It follows that X must be locally modular. As G is almost strongly minimal, G is a 1-based group. By [HP], and the "minimality" of G, G is already strongly minimal. Let q be the generic type of G. Proposition 4.10. The above relationship establishes a bijection between the nonorthogonality classes of nontrivial Morley rank 1 types which are orthogonal to PC? and the isogeny classes of simple abelian varieties which do not "descend" toGu Proof. Let A, B be simple abelian varieties, G, H strongly minimal locally modular definable subgroups of A, B respectively. Let g, r be the generic types of G, H respectively. From the above discussion it is clearly enough to prove that A is isogenous to B iff q is nonorthogonal to r. First suppose that f is a rational isogeny of A with B. Thus /(G) is a strongly minimal subgroup of B. By 4.6, /(G) = H. Thus (as / is finite-to-one), / witnesses the nonorthogonality of q and r. Conversely suppose q is nonorthogonal to r. As G and H are both locally modular strongly minimal groups it follows that G and H are definably isogenous. (After naming enough parameters, we have generics α £ G, b £ H such that a and 6 are interalgebraic. Now G x H is a "1-based group". Thus by [HP] tp(a,b) is the generic type of a strongly minimal subgroup of G x H, which yields the required isogeny. By 4.7 A and B are rationally isogenous.

133

§5 Zariski-dense definable subgroups of simple algebraic groups. Finally we give a sketch proof (due essentially to Buium [B2]) of the infinite Morley rank case of Cassidy's theorem. We start with a simple (noncommutative) group G, definable in U and of infinite Morley rank. The aim is to show that G is definably isomorphic to an algebraic matrix group H C GL(n, U) for some n. First, a proof like that of Corollary 1.2, but using so-called *-groups, shows that G is definably embeddable in GL(n, U) for some n. (This appears in [P2].) Let H be the Zariski-closure of G. Using the simplicity of G, we may assume that H is simple (i.e. has no normal algebraic subgroups). So we are finished if we prove: Proposition 5.1. Let if be a simple algebraic group over U, and G a Zariskidense definable (in U) subgroup of H with infinite Morley rank (in U). Then G=H. Proof. It is known that any simple algebraic group over an algebraically closed field is rationally isomorphic to a matrix group defined over the prime field (namely Q). So we may assume H is such. We make use of the following: Fact. The action of H on its Lie algebra by the adjoint representation is irreducible. Let us go back to the "jet groups" HI, HZ, etc. As in the beginning example in section 3, the action of H on fcer(ττι), defined by : for α G H, 6 G fcer(τrι), let αi G H\ be such that ττι(αl) = α, and define 6α = αj"1 6 α, is exactly the adjoint action of H on Lie(H). It follows similarly that for any m, the action of Hm on ker(πm+ι), defined in a similar fashion, is isomorphic to the action of H on Lie(H). Thus all this actions are irreducible. Now suppose that G φ H. By Lemma 3.4, for some m, Gm is a proper algebraic subgroup of Hm. Choose least such m. (Note m > 0, as GO = HQ = H). It follows that τr m |Gm : Gm -+ Hm-ι is surjective. Let Lm C Hm be the kernel of ττ m . Thus Lm Π Gm is a proper subgroup of Lm which is invariant under the above-mentioned action of Hm_\. By irreducibility, Lm Π Gm = {0}. This says that ττm|Gm is finite-to one. In particular for generic α G G,α<m) Gacl(α,α', ...,aίm-l\k) (where k is some field over which G is defined). It follows easily that RM(tp(a/k)) is finite, thus RM(G) < ω, contradiction.

134

References [Bo] A. Borel, Linear Algebraic Groups, Springer 1991. [Bl] A Buium, Differential Algebraic Groups of Finite Dimension, Springer Lecture Notes 1506, 1992. [B2] A Buium, Differential polynomial functions on algebraic varieties I: Differential algebraic groups, American Journal of Mathematics, 1993. [Cl] Ph. Cassidy, Differential algebraic groups, American Journal of Mathematics, 94 (1972), 891-954. [C2] Ph. Cassidy, The classification of semisimple differential algebraic groups, J. Algebra, 121 (1990), 169-238. [H] E. Hrushovski, Locally modular regular types, in Classification Theory, ed. J. Baldwin, Lecture Notes in Math. 1292, 1987. [HP] E. Hrushovski and A. Pillay, Weakly normal groups, Logic Colloquium '85, North-Holland, 1987. [HS] E. Hrushovski and Z. Sokolovic, Minimal subsets of differentially closed fields, preprint 1994. [HZ] E. Hrushovski and B. Zilber, Zariski geometries, to appear in Journal of A.M.S.. [L] S. Lang, Introduction to Algebraic Geometry, Interscience, New York, 1959. [Las] D. Lascar, Stability in Model Theory, Longman, 1987. [M] Yu. Manin, Rational points of algebraic curves over function fields, AMS Translations, Ser. II 50 (1966), 189-234. [Mr] D. Marker, Model theory of differential fields, this volume. [PI] A. Pillay, Differential algebraic group chunks, Journal of Symbolic Logic, 55 (1990), 1138-1142. [P2] A. Pillay, Some foundational questions concerning differential algebraic groups, preprint 1994. [P3] A. Pillay, Geometrical stability theory, to appear, Oxford University Press. [Po] B. Poizat, Groupes Stables, Nur al-Mantiq wal-Ma'rifah, Paris 1987. [R] M. Rosenlicht, Extensions of vector groups by abelian varieties, American Journal of Mathematics, 80 (1958), 685-714. [Sh] I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1977. [SHM] S. Shelah, L. Harrington and M. Makkai, A proof of Vaught's conjecture for ω-stable theories, Israel Journal of Mathematics, 49 (1984), 259-278. [Si] J. Silverman, Arithmetic of Elliptic Curves, Springer, 1987

Some model theory of separably closed fields Margit Messmer

§1 Introduction. The model theory of separably closed fields was first investigated by Ersov. Among other things he proved that the first-order theory of separably closed fields of a fixed characteristic p / 0 and of fixed degree of imperfection e £ ω U {00} is complete, see [6]. In 1979 C. Wood (see [24]) showed that these theories are stable, but not superstable, yielding the only examples of stable, non-superstable fields. Further model theoretic properties of these fields, like quantifier elimination, equationality, the independence relation, DOP, etc. were analysed. In 1988 F. Delon (see [5]) published a comprehensive article in which she investigated types in terms of their associated ideals in an appropriate polynomial ring, in particular proving elimination of imaginaries and giving a detailed analysis of different notions of rank. In 1992, E. Hrushovski gave a model theoretic proof of the Mordell-Lang conjecture for function fields. In the case of characteristic p φ 0 he used some of the model theoretic tools for separably closed fields, in particular an analysis of minimal types and the author's results on definability in separably closed fields. A separably closed field can be equipped with a differential structure. Accounts of this line of work can be found in [21,22,23,10]. The purpose of these notes is to give an overview of the known results in the model theory of separably closed fields with special emphasis on the case of finite degree of imperfection. When discussing elimination of imaginaries, we give a general outline of how this property can be proved in all known examples of stable fields with additional structure

§2 A few remarks about field extensions. All fields under consideration in this chapter will be of fixed characteristic p φ 0. F, K and L always denote fiejds, F[Xi,i £ /] the polynomial ring over F in the indeterminates Xi,i £ /. F stands for the (field-theoretic) algebraic closure of F and Fp* for the subfield {xp" : x G F}. By abuse of notation Fn denotes the set of n-tuples over F. To avoid confusion, we will sometimes use [F]n for the cartesian product. fp is the finite field with p elements.

136

Definition 2.1. A polynomial / £ F[X] is said to be separable if all_its irreducible factors (in F[X]) have distinct roots (in F). An element x £ F is said to be separable over F if its minimal polynomial over F is separable. Note: An irreducible polynomial / £ F[X] is separable iff its formal derivative /' is nonzero. An algebraic extension K of F is separable if every x £ K is separable over F. Now let us define what it means for an arbitrary field extension to be separable. Definition 2.2* (a) A p-monomial over a set {αi,... ,α n } C F is an element of the form a\l " α£n with 0 < e, < p. A finite set A — {αι,...,α n } C F is p-independeni in F if the set of p-monomials {ra0 = \^..,mp^_ι} over A is linearly independent over Fp. An infinite set is p-independent if every finite subset is. (b) A field K D F is a separable extension of F if, whenever A C F is p-independent in F, then A is p-independent in /f. (c) A set A C F is a p-basis of F if the set of p-monomials over A form a basis for F over Fp as a vector space; i.e. A is a maximal p-independent subset of F. The cardinality of such a set A is called the degree of imperfection or Ersov-invariant of F. We will simply call it the invariant of F . (d) F is said to be separably closed if it has no proper separable algebraic extension. F denotes the separable closure of F, that is the maximal separable algebraic extension of F (inside F). Note: • Part (b) of the previous definition is just another way of saying that F and Kp are linearly disjoint over Fp. • The property of being separably closed can be expressed by an infinite set of first-order sentences in the language C = {+, —, , ~ 1 ,0,1} of fields by saying that each polynomial whose formal derivative is nonzero has a root.

§3 The theory of separably closed fields in the language of fields. By the note at the end of the previous section we can form the first-order theory SCFe of separably closed fields (of characteristic p) of invariant e, where e £ ω or e = oo in the language C = {+,-, , -1 ,0,1} of fields. Notice that SCFo is the theory of algebraically closed fields. First we show that SCFe is complete.

137

We first consider the case when e is finite. We extend the language £ by finitely many constant symbols α i , . . . , α e interpreted as the elements of a pbasis in each model of SCFe. So £' = £ U {αi, . . . , α e } and SCF'e stands for the theory of separably closed fields of invariant e in the language £'. It is clear that SCFe and SCF'e have the 'same' models and completeness of SCF'e implies the completeness of SCFe . Definition 3.1. A theory T is model complete if for all models M , N of T, M C N implies M ^ N. Note: T is model complete iff for all models M,N of T and every existential sentence 3xφ(x,m) with ra C M and ^ quantifier-free, if TV [= Ξx<£(x,ra) then M [= Lemma 3.2. The theory SCFg is model complete. Before proving the lemma we make a few remarks about varieties (in the sense of Lang, see [8]). Let Ω be an algebraically closed field. By an (affine) variety V we mean the zero set of a prime ideal T> of Ω[ΛΊ, . . . ,X n ] for some n\ that is V = {x G Ωn : f ( x ) = 0 for all / G P}. Conversely, the ideal I(V) associated to a variety V is given by {/ G Ω[X] : f ( x ) = 0 for all x G V}. Let / be an ideal of Ω[X]. If 7 has a basis consisting of elements from K[X] with ^ C Ω, then K is said to be a /ίe/d o/ definition of /. (We will note later that there exists a minimum such field of definition, see Section 4.) The variety V is said to be defined over K if K is a field of definition for I(V). Lemma 3.3. Let F be separably closed and K a separable extension of F. Then F and K are linearly disjoint over F. Proof. Let {&ι,...,δ n } C if be linearly dependent over F. So there are ci, . . . , cn G -F, not all zero, with CIOH-----\-cnbn = 0. Since F is separably closed, each Cj is purely inseparable over F. So there is ra G ω such that cf G F for all i. So we have cξ 6f H-----hc^™^™ = 0. Now, since /£ is separable over F, if follows that X pm and F are linearly disjoint over Fpm . Therefore we find dι , . . . , dn G F, not all zero, with d^b^ + + d£m6£m = (dι&ι + + dnbn)pm = 0, which says that {61, . . . , bm} is linearly dependent over F. Proof of 3.2. We follow Ersov's proof, see [6] and also [24, Th.l]. Let F C K, both models of SCF^. Since {βι,...,α e } is a p-basis of both F and /C, it follows that K is a separable extension of F (p-independence is preserved!). Furthermore, since F is separably closed, F is relatively algebraically closed in K. So K is a 'regular' extension of F. Now let φ(xιt. . . , z n ) be a quantifier free

138 formula over F with K \= 3xφ(x). Let 6 C K with K |= φ(b). Without loss of generahty φ is in disjunctive normal form; that is

Φ = \ΛΛ £«•(*) = ° Λ Λ «*(*) * °) with fji, gjk € F[X]. Without loss of generality K (= Λi /«(&) = ° Λ Λ 0. Now consider the prime ideal P C F[X,Y]

defined as follows.

By the previous lemma, F and K are linearly disjoint over Fλ so by [8,Ch.IΠ, Th.8] , F is a field of definition of P and the variety V C Fn+1 given by P is defined over F. By [8,Ch.IΠ, Th. 10], the set of points of V which are separably algebraic over F is dense in V '. Since F is separably closed, there is (c, d)eVΠ F"*1. Clearly, since fa(X) E P for all i and Π* 0ι*(*)y - 1 € P, c satisfies φ(x). Remark 3.4. The separable closure of FP(Q.\, . . . ,α e ) is the prime model of Proof. Clearly the separable closure of Tp(a\^ . . ,α e ) is a model of and it is contained in any model of SCF'e. The claim follows from the model completeness of SCF'e. Theorem 3.5. (Ersov) The theory SCFe (eeωU {oo}) is complete. Proof. For e finite, Lemma 3.2 and Remark 3.4 show that SCF'e is complete, which implies the completeness of SCFe . In the case of e = oo, instead of adding constant symbols for a p-basis we add infinitely many relation symbols Qn(xι, >^n), n G ω, interpreted as follows: Qn(^ij •• ,ZM) iff {zι, £n} isp— independent. So

Q n (zι,...,z n ) iff V y i . . . yn(yplχι + '-y*χn = Q-+yι = - - = yn = 0). In this extended language one can show model completeness in a similar way as before. Again, the separable closure of fp(Xi : i G ω) is the prime model which yields the completeness of SCF^ . (For the infinite invariant part of this proof seealso[24Th.l] .)

139

Note: (a) The theory SCFe (in the language of fields) is not model complete: Let F < K (= SCFe, where K = F(b) with V> = a G F-F*>. Then K |= 3φ*> = α), but F £ Ξz(z*> = α). (b) In the case of infinite invariant it is not possible to work in a language where we have names for a p-basis, since elementary extensions can contain new p-independent elements.

§4 Separably closed fields of finite invariant. In the previous section we saw that the theory of separably closed fields of finite invariant is model complete if we add names for a p-basis. After extending this language by some definable function symbols, this theory will turn out to have several 'nice' model theoretic properties, such as quantifier elimination. From now on we fix e to be finite and nonzero. Let F |= SCF'e and let {mo = l,...,m p e _ι} be the set of p-monomials over the p-basis {αi,..., ae} as before. Each element of x G F can uniquely be written of the form x =2

with X(i) G F. Now for 0 < i < pe, W = *M)mO + χP(i

X

Continuing this process, we get a tree associated to each element x G F:

Note that in the language £' each element in the tree is definable over x, and that x is definable over each level of the tree. Let (pe)<ω denote the set of finite tuples over the set {0, . . . ,pe — 1}. We extend the language C! by infinitely many unary function symbols λ σ ,σ G (pe)<ω , interpreted as follows:

• For 0 < j < pe, \(j)(x) = X(j) iff x = Ejlό1 χ P ( j ) m j • For σ G (pe)<ω, λσ(j)(x} = λ(j)(λσ(x))

= Xσ(j), as indicated in the tree.

140 We say that x^ ) is the jth coordinate of x. Note that all λ σ 's are definable in the language C! '. Definition 4.1. SCF* denotes the theory of separably closed fields (of char1 acteristic p) of invariant e (< ω) in the language C* - {+,-, » " > 0 , 1} U {al,...,ae}U{λσ:σe(p*)<»}. Note that by 3.2 and 3.5 SCF* is model complete and complete. Proposition 4~.2. The theory SCF* eliminates quantifiers. Note: Delon in [5, Prop. 27] discusses a more general language for the finite and infinite invariant yielding quantifier elimination. Proof of Proposition 4.2. In order to prove quantifier elimination for a theory T it suffices to show the following (Shoenfield test): For any ωι-saturated model M of T and any countable model N of T, and substructures A C TV and B C M, if A = B then this isomorphism extends to an elementary embedding of N into M . Claim. Let F (= SCF* and k a substructure of F, then k has invariant e. Proof of Claim. Clearly the invariant of k is at least e since {αi, . . . , α e } C k. But with any element x £ fc, k contains all coordinates xσ of x witnessing its p-dependence on {αi, . . . ,α e }. This shows the invariant of k is at most e. This proves the claim. Now let kι be a substructure of K \= SCF* with K ^-saturated and k^ a substructure of F |= SCF* with F countable, and Jbi Ξ Jk2 Clearly fci 2 ίc2 with ki \= SCF* and k\ C K. By model completeness k\ -< K and k^ embeds elementarily into K. But since K is u>ι -saturated, F also embeds elementarily into K. This quantifier elimination results yields a very useful one-to-one correspondence between 1-types over models of SCF* and certain prime ideals in a suitable polynomial ring. This feature is explored in great detail in [5]. Here we discuss a few aspects of it. Corollary 4.3. In SCF* every formula φ(x) with parameters from a model F \= SCF* is equivalent to a boolean combination of formulas of the form

where / € F[Xir , . . . , Xnσ : σ E (p*)<ω}.

141

Proof. First check that for all x,y and σ G (pe)<ω ', (z + y)σ, (x - 2/)<τ, (z y)σι (x~l)σ £ F(xτ,yτ : T G (p € ) < ω )< For example, for p = 3, e = 1 and p-basis {α}, the first-level coordinates of x - y can be obtained as the first row of the matrix

2/{ι)α Now the claim follows immediately from Proposition 4.2. The following remark shows that all information about types is contained in the 1-types. Remark 4.4. Let F |= SCF%. There is a definable injection from [F]n into F, and therefore from the set of ra-types into the set of 1-types. Moreover, if n = pm'e for some m < ω, we get a bijection. Proof. Let m < ω be such that pm'e > n. Let {m^, . . .,m^ me } be the set of pm -monomials over {αi, . . . , αe}; that is {m^, . . . , m^me} is a basis of F over F*>m. Define Φ : [F]n -> F by

«=ι

SoΦ(x) is an element whose mth level of its tree consists of (xi, . . . ,x n ,0, . . . ,0). Φ is the desired injection. For F \= SCF*y the automorphisms Aut(F) of F act on the types over F by acting on the parameters. For F[Xi : i G J], a polynomial ring over F, Aiίί(F) also acts on the ideals of F[Xi : i G /] by acting on the coefficients of the polynomials. Corollary 4.5. Let F \= SCF* . There is a (natural) one-to-one correspondence between complete 1-types over F and 'certain' prime ideals in the polynomial ring F[Xσ : σ G (pe)<ω]5 given as follows. Let q be a 1-type over Fy then the ideal I(q) associated to q is given by = {fe

F(Xσ]σ

e

G (p )<Ί •' 7(*σ) = 0' G q}.

Moreover, any automorphism α G Aut(F) fixes q (setwise) iff α fixes I(q) (setwise). Proof. Immediate from Corollary 4.3. Note that I(q) is a prime ideal since q is a complete type.

142 Note: • (Delon) We call an ideal / of F(Xσ\σ e :σ£(p )<»],

e <ω

G (p ) ]

separable if for all

β

/£ _ι™p -ι G / implies / 0 ,...,/ p e-ι G /. The Certain' ideals occurring as ideals of types are exactly the prime ideals which are separable in this sense and contain

«=o • (Delon) The same one-to-one-correspondence holds for types over definably closed sets, see [5, Prop.33] . • This kind of description of types in terms of ideals also arises in algebraically closed fields and in differentially closed fields. Corollary 4.6. (Wood) The theory SCF* is stable, not superstable. Proof. Let F \= SCF* with \F\*° = \F\. By Corollary 4.5 the number of 1-types over F is at most the number of ideals of F[Xσ\σ G (pe)<ω] For an ideal ICF[Xσ]σe(pe)<ω]\et

where (pe)n denotes the set of tuples over {0,... ,pe — 1} of length at most n. F[Xσ : σ G (pe)n] is a noetherian ring, since it is a polynomial ring with finitely many indeterminates. Therefore In is finitely generated. So there are at most |FI possible different In for each n. But / = U neω ^n, which shows that there at most |F|K° = \F\ possible different ideals /. To see that SCF* is not superstable, we show that

F > Fp > Fp2 > - - > Fpt >

-

forms an infinite descending chain of definable additive subgroups, each of infinite index in the preceding one. (The same can be shown for the corresponding multiplicative subgroups.) Since F is definably isomorphic to Fp via the Frobenius map x H-> xp, it suffices to show that the index of Fp in F is infinite. But this follows from the following theorem of Poizat's, see [12, Th.5.10]. Let F be a stable field. The F has no definable (additive or multiplicative) subgroup of finite index. We can also see this directly. Let α G F - Fp and 6, c G F* = F - {0} with b φ c. at? and acp lie in different (additive) cosets modulo Fp, since α&P — ac? = dp implies a = (ι^)p, a contradiction.

143

This corollary also shows that the theories SCFe and SCF'G from Section 3 are stable, not superstable for t E ω. The same holds for SCF^, see [24, Th.3] and [5, p.63]. Note. Separably closed fields are the only known stable, non-superstable fields. Next we want to list several model-theoretic properties of the theory SCF*. 1. Quantifier elimination implies that SCF* is an equational theory for e < ω. This was shown by G.Srour, see [17]. It is not known whether SCF<χ> is equational. 2. Result 5.10 together with 4.5 and 4.6 show that the theory SCF* of separably closed fields with e < ω eliminates imaginaries. (For definitions, etc. see Section 5) 3. OOP ('The Dimensional Order Property*) is a non-structure property which for a superstable theory T yields the maximal number of models of in each cardinality > 2'TL In [1], Bouscaren proved that every superstable theory with a stable theory of pairs does not have DOP. Subsequently Delon showed that separably closed fields provide an example of a stable theory with stable pairs which has DOP. In fact, in [3] a strengthened, infinitary version, called ω-DOP, is proved. The authors show that there is a family of independent models /f, , i E ω, each containing a fixed model KQ , and a type p over the prime model over UiKi such that for all j E ω, p is orthogonal to U^jKi. Among other things, the proof makes use of the fact that there is an algebraic description of (in)dependence, which we want to mention here. 4. Forking in SCF* Fact 4.7. Let F C K be models of SCF* and p a complete type over K realized by some x£L>K. Then p does not fork over F iff F(xσ : σ E (pe)<ω) and K are algebraically disjoint over F. This is equivalent to saying that F(xσ : σ E (pe)<ω) and -K" are linearly disjoint over F. For proofs and further details see [5, p.Slff.]. This description of forking says that two elements x and y are modeltheoretically independent over some model F of SCF* iff their trees {xσ : σ E (pe)<ω} and {Vσ '• & G (pe)<ω} are algebraically independent over F. Similar descriptions of independence we find in algebraically closed fields and differentially closed fields. 5. Non-FCP. The theory SCF* does not have the 'finite cover property' (nonFCP). The proof can be copied from the Chapter 'Model Theory of Differential Fields' by D. Marker, replacing differential polynomials there by polynomials in the polynomial ring F[X\σ,... ,Xnσ •' & € (pe)<ω] There an explicit proof of 'uniform bounding' is given. Non-FCP follows by Shelah's FCP-Theorem

144

(see [16, Ch.II, Th.2.2(8)]) using elimination of imaginaries and the stability of

SCF*. 6. Groups and fields. (For details see [9]) As in algebraically closed fields and differentially closed fields the question arises of whether the groups definable in a separably closed field are related to 'classical' groups. Results by Weil, van den Dries and Hrushovski showed that any infinite group definable in an algebraically closed field is definably isomorphic to an algebraic group. For a proof of this result see [2]. In analogy to (abstract) algebraic groups (over algebraically closed fields) we introduce the notion of an F-algebraic group for a separably closed field F^SCF*. Definition 4.9. A subset A C [F]n is called F -closed if there are polynomials /i, i fm G F [Xi, . . . , Xn] , m G ω such that A = {x G [F]n : f i ( x ) = 0 for i = 1, . . . , m} .

We call V = V\ U . . . U Vk a variety in F if there are F-closed 'charts' Ui C [F]n and bijections /< : Vi -> Ui such that Uij = fi(Vi Π Vj) are F-open subsets of Ui and such that the fa = fj o /, -ι : Uij -* Uji are rational functions over F, for 1 < i y j < k. We call (G, •) an F -algebraic group if G is a variety in F such that the maps (x,y) *—>• x - y and x ι-» a:""1 are morphisms with respect to the topology defined above, i.e. are locally F-rational functions. Prop 4.9. Every infinite group G interpretable in a model F of SCF* is definably isomorphic to an F-algebraic group. n

Sketch of proof. A natural topology on F is the λ-topology given as follows. n

A subset A C F is called basic \-closed if there are finitely many e <ω polynomials /,- G F [Xισί. . . ,Xn
145 1

1

independent generic elements of G '. Then we can cover G by finitely many translates of a λ-open generic' subset on which multiplication and inversion 1 are rational functions, so that G is equipped with the structure of a λ-alebraic group. Finally the following fact allows us to turn this λ-algebraic group into an F- algebraic group. Let A be a λ-closed subset of F \= SCF* defined by a formula of the e <ω form f(xσι, . . . ,x σ j = 0 with / G F[Xσ\σ G (p ) ]. Then A is in m definable bijection with an F-closed subset B of [F] for some m G ω defined by g(x) = 0 for some g G F[Xι, - - - Xm]To see this, first let L G ω be occuring in /. We can assume that the xσt by the corresponding term Le m = p and define the map Φ : F

the maximal length of all tuples σi, . . . , σn all σ, are of the same length L, by replacing in the x r 's, where τ has length L. Now let m —> [F] as follows.

where {τi,...,r m } are all the tuples in (pe)<ω of length L, or equivalently (xTl , . . . , x T m ) is the Lth level of the tree of x. Clearly the image B of A under Φ is defined by the same polynomial / viewed as an element of F[Xι , . . . , Xm]. As a corollary of the previous result one can prove the following Rosenlichtstyle theorem (see [15]) for infinite groups interpretable in SCF*. Corollary 4.10. Let G be a connected infinite group interpretable in a separably closed field F (= SCF*. Then G/Z(G) is definably isomorphic to a linear Falgebraic group. (Z(G) denotes the center of G.) Note: By a linear F-algebraic group we mean an F-closed subgroup of GL v(F) for some TV < ω, the general linear group over F . Now we can classify the infinite fields interpretable in SCF* . Theorem 4.11. Let K be a field interpretable in a separably closed field F \= SCF* of finite invariant. Then K is definably isomorphic to a finite (purely inseparable) extension of F. In particular, K is itself separably closed. Sketch of Proof. By Corollary 4.10 the additive group K+ as well as the multiplicative group K* are both linear F-algebraic groups which are the F-rational points of some linear algebraic groups F"1" and V* , respectively, in the algebraic closure F of F. Since K* acts on K+ by multiplication, one can show that V* acts on V+ . After restricting to the semisimple part of V* , we are in the situation to apply ZiΓber's field theorem (see [12, Theorem 3.7]) and find an algebraically closed field 1C definable in F which is definably isomorphic to F by [12, Theorem 4.15]

146

, via a definable isomorphism Φ. We arranged that K be definably embedded into 1C. So Φ carries K onto a subfield of some finite extension of F. But it pn p is easily seen that such a subfield must contain F for some n. Now F " is isomorphic to F itself, so K is definably isomorphic to a finite extension of F. Using an ultraproduct argument, it can also be shown that any infinite field K definable in a separably closed field F of infinite invariant is itself separably closed of infinite invariant, and char(K) — char(F). As a corollary of the previous theorem, Hrushovski observed the following. Note that in a saturated model, a set X, which is defined by a possibly infinite conjunction of formulas, is called minima/if for every definable (with parameters) set A, the intersection of X with A is finite or cofinite in X. Corollary 4.12. Let F be a separably closed field of finite invariant and K an infinite minimal field defined in F by an infinite conjunction of formulas. Then K is definably isomorphic to Fp°° = Γ\nFpn, the maximal perfect, algebraically closed subfield of F. Proof. By [12, Cor.5.21] K is the intersection of fields Li each definable in F. By Proposition 4.11, each Li is definably isomorphic to a finite extension Ki of F. By applying the map x ι—»• xp for some large enough n to the Ki's, we get an infinite descending chain of definable subfields K[ of F. By [9, Cor.3.2], each Kl contains Fp°°. But since K is minimal, Γ(K[ = Fp°°.

§5 Elimination of imaginaries in fields. In model theory we often find structures which are given on definable sets modulo some definable equivalence relation, for example a definable group modulo some definable normal subgroup or projective space over some field, etc. These structures are, in a strict sense, not definable. Often one says that they eq are interpretable in the theory T , or definable in T . If a theory T eliminates imaginaries, each such interpretable structure is definably isomorphic to an (honestly) definable structure (see Fact 5.2(a) below). We will explore this property in the case of stable fields. For a general discussion of elimination of imaginaries, see for example [13]. Most of the discussion here will appear in [10]. Definition 5.1. A first-order theory T with monster model M is said to eliminate imaginaries if for every definable (with parameters) set A C Mn, there is a finite set B C M such that for every automorphism σ of M> σ fixes A setwise if and only if σ fixes B pointwise.

147 Fact 5.2. (a) If the theory T eliminates imaginaries and the definable closure (=dcl) of the empty set contains at least two elements, then for every definable equivalence relation E on Λ4 n there is a definable function / from Mn to M.m for some m such that for all z, y, E(x, y) iff f ( x ) = f ( y ) (b) If the theory T eliminates imaginaries then for every definable set A C Mn there is a minimum definably closed set B such that A is definable over B. Proof. For (a) see [13, Th.16.16] . (b) is immediate from the definition. Remark 5.3. The theory SCFe for e G ωU{oo} (in the language of fields) does not eliminate imaginaries.

Proof. (See [5]) Let F be a saturated model of SCFe. Then we find can two elements x, y in F — Fp which lie in the same (multiplicative) coset modulo F* and such that Fp(x)Γ\Γp(y) = Tp. Now fp(x) and Fp(y) are two definably closed sets over which the coset xF* is definable. But clearly xF*P is not definable over Fp(x) Π Fp(y] = Tp, contradicting Fact 5.2(b). At this point we would like to pick the example from the previous proof and show how the quotient group F*/F*P can be eliminated in the theory SCF*. We consider the specific example when p = 3 and e = I with the p-basis being {α}. The two elements x, y E F* lie in the same coset modulo F* iff

x

Note that x = x?0v + x^n\a + x?2)α2' implies that

y an(

^ sinularly f°Γ V

The equation above

which implies that - G ^* - So the definable closure of

is the minimum definably closed set over which xF* is definable. Or in terms of Fact 4.8(a), the quotient group F*/F* can be eliminated by the definable map x3(1)+2x3(2)a JU I—T

.

X

There is a weaker version of elimination of imaginaries.

148

Definition 5.4. A theory T with monster model M has weak elimination of n imaginaries if for every definable set A C M there is a formula φ(x,y) such that there are only finitely many tuples ά i , . . . , άm such that φ(x, α, ) defines A. Note that a theory has elimination of imaginaries iff for every definable set n A C M there is a formula φ(x, y) such that there is a unique tuple a such that φ(x,a) defines A. In the theory of fields these two definitions of elimination of imaginaries are equivalent. Fact 5.5. Let T be the theory of a field. Then T has weak elimination of imaginaries iff T has elimination of imaginaries. Proof. See [14, Cor.6]. Let φ(x,y) be a formula in the language of T. We give the idea of the proof in the case where y has length one. Let α i , . . . , αm be the only elements such that φ(x, αt ) defines A C Mn. Let /ι(yι,..., y m ),... , /m(yι, - - , ym) be the symmetric functions in yι,..., y m ; that is

Now define the formula ψ(x,z) to be m

m

*> yi) Λ /\ Vz0(z, yi) <-> <£(*, y, ) Λ /\ y, / y; Λ /\ zf = /< Now (/ι(α), . . . , /m(α)) is the only tuple 6 such that ψ(x, b) defines A. The proof is based on the fact that using symmetric functions, we can code up finite sets as finite tuples. In the general case when y has length /, one can code the set {α~Ί, . . . , α^} by the tuple with consists of the coefficients of m

J|(y + OiiXi + ai2X2 +

+ Oi

Definition 5.6. Let T be an arbitrary first-order theory and p an n-type over a model M of T. A definably closed set A C M is said to be the canonical base ofp if for every automorphism σ of M, σ fixes p iff σ fixes A pointwise. To point out the connection between elimination of imaginaries and the existence of canonical basis we include the following lemma.

149

Lemma 5.7. Let T be a stable theory with elimination of imaginaries. Then every type has a canonical base. Proof. Let p be an n-type over a model M of T. Then p is definable over M. This means that for every formula φ(x, y) over the empty set there is a formula dφ(y) over M such that for all a C M, φ(x, a) £ p iff M (= dφ(ά). Since T eliminates imaginaries, for each dφ(y) there is a finite set £^ such that every automorphism fixes the set defined by dφ(y) setwise iff it fixes B pointwise. Now let A be the definable closure of the union of all Bφ. It is easy to check that A is the canonical base for p. The converse of the previous Lemma is not true. The theory of an infinite set in the pure language of equality is a counterexample. But we have the following. Proposition 5.8. (Evans, Pillay, Poizat, see [7]) Let T be a stable theory such that each n-type over M has a canonical base for every model M of T. Then T has weak elimination of imaginaries. Proof. Let A C ΛΊ n be a set defined by φ(x,a) and let E be the following equivalence relation. yEziff Mxφ(x,y)^φ(x,z). Furthermore let C = {y : Vxφ(x, y) <-»• φ(x, a)} = the class of ά.

Pick p, a nonforking extension of V>(y, ά) to Λί, and let B be the canonical base of p. Note that each (τ(B) gives rise to a nonforking extension of ^(y,ά) to Λ4, of which there is only a bounded number. Thus B has only a bounded number of images under automorphisms σ which preserve C. Claim 1. C is defined over B. Proof of Claim 1. Let σ be an automorphism of M fixing B. So σ fixes p and ^>(t/, <τ(ά)) £ p, defining the equivalence class <τ(C) of E. Since equivalence classes are disjoint, it follows that σ(C) = C. Thus each automorphism fixing B fixes C. Hence C is defined over B by some formula 0(i/,6), b C 5, proving the claim. Claim 2. There is only a finite number of &' with the same type as 6 over the empty set such that 0(i/,6) <-»• θ(y, 6'). Proof of Claim 2. If σ is an automorphism_of M with σ(6) = 6' such that 0(y>&) «-* θ(y>tf)> then_σ preserves C. Since 6 C 5, there is only a bounded number of such images 6' by the note above. So by compactness there is only a finite number of 6"s. This proves the claim.

150 Now again by compactness, there is a formula χ ( z ) in the type of 6 over the empty set which implies the statement of claim 2. Define

Then the formula

φ*(yι,z)^ has the property that there are only finitely many 6's such that Γ(z,6) defines A. Corollary 5.9. Let T be the theory of a stable field such that for each model M of T each n-type over M has a canonical base. Then T has elimination of imaginaries. Proof. By 5.8 and 5.5. As mentioned before, there are several examples of stable fields (with additional structure) where we find a one-to-one correspondence between complete n-types and certain ideals in an appropriate polynomial ring. This immediately yields the existence of canonical bases as follows. Let F be a field and F[Xj : j £ J] a polynomial ring over F. In Section 3 we discussed the notion of a 'field of definition' of an ideal I of F[Xj : j £ J]. In fact every such ideal / has a minimum field of definition C(I) contained in any other field of definition and obtained in the following way. Let M be the set of monomials over {Xj : j £ J}. Then there is a subset MO of M which is a basis for F[Xj : j £ J]/I- So modulo /, every monomial πik £ M can uniquely be written as Σι α fc/ m / w^h αjbj E F, mi G MQ. Then C(I) is the subfield of F generated by all the α*/ Moreover, C(I) has the property that for any automorphism σ of F, σ fixes / (setwise) iff σ fixes (7(7) pointwise. For proofs see for example [8]. Now we can formulate a general recipe for proving elimination of imaginaries in certain stable fields. Prop 5.10. Let T be the theory of a stable field. Suppose that for every n> 1 there is a (possibly infinite) set of indeterminates Xi, i £ J, such that for each model F of T there is a one-to-one correspondence between complete n-types over F and certain ideals in the polynomial ring F[Xi : i £ J], such that for every automorphism σ of F (as a T-structure) , σ fixes the type (setwise) iff σ fixes the corresponding ideal (setwise). Then T eliminates imaginaries. Proof. For any n-type p over F let I(p) be the corresponding ideal. Then the definable closure of its minimum field of definition C(I(p)) is the canonical base of p. Therefore, by Corollary 5.9, T eliminates imaginaries.

151

References [I] E. Bouscaren, Dimensional order property and pairs of models. These d'etat. [2] E. Bouscaren, Model-theoretic versions of WeiVs theorem on pregroups. The Model Theory of Groups. Ed. A.Nesin and A.Pillay. University of Notre Dame Press, 1989, pp.177-185. [3] Z. Chatzidakis, Cherlin G., Shelah S., Srour G., Wood C. Orthogonality in separably closed fields. Classification Theory. Ed. J. Baldwin. New York, Springer, 1985, pp. 72-88. [4] G. Cherlin and S. Shelah, Superstate fields and groups. Annals of Mathematical Logic, vol.18 (1980), pp.227-270. [5] F. Delon, Ideaux et types sur les corps separablement clos. Supplement au Bulletin de la societe Mathematique de France. Memoire No.33. Tome 116 (1988). [6] J. Ersov, Fields with a solvable theory. Doklady Akadeemii Nauk SSSR, vol.174 (1967), pp.19-20; English translation, Soviet Mathematics, vol.8 (1967), pp.575-576. [7] D. Evans, A. Pillay, B. Poizat, A group in a group Algebra i Logika, No.3 (1990), pp.368-378. [8] S. Lang, Introduction to algebraic geometry. Interscience publisher, New York, 1958. [9] M. Messmer, Groups and fields interpretable in separably closed fields. Transactions of the AMS, vol.344 (1994), pp. 361-377. [10]M. Messmer and C. Wood, Separably closed fields with higher derivations I. Journal of Symbolic Logic, to appear. [II] A. Pillay, An Introduction to stability theory. Clarendon Press, Oxford, 1983. [12] B. Poizat, Groupes stables. Nur alMantiq walMa'arifah, Villeurbanne,1987. [13] B. Poizat, Cours de theories des modeles. Nur almantiq walMa'arifah, Villeurbanne, 1985. [14] B. Poizat, line theorie de Galois imaginaire. The Journal of Symbolic Logic, vol.48 (1983), pp.1151-1171. [15] M. Rosenlicht, Some basic theorems on algebraic groups. American Journal of Mathematics, vol.78 (1956), pp.401-443. [16] S. Shelah, Classification theory and the number of non-isomorphic models. North-Holland, 2nd ed., 1990. [17] G. Srour, The independence relation in separably closed fields. The Journal of Symbolic Logic, vol.51 (1986), pp.715-725. [18] L.P.D. Van den Dries, Definable groups in characteristic 0 are algebraic groups. Abstracts of the American Mathematical Society, vol.3 (1982), p.142.

152 [19] L.P.D. Van den Dries, Weil's group chunk theorem: a topological setting. Illinois Journal of Mathematicsx vol.34 (1990), pp.127-139. [20] Weil,A. On algebraic groups of transformations. American Journal of Mathematics, vol.77 (1955), pp.203-271. [21] C. Wood, The model theory of differential fields of characteristic p φ 0. Proceedings of the AMS, Vol.40 (1973), pp.577-584. [22] C. Wood, Prime model extensions for differential fields of characteristic p φ 0. The Journal of Symbolic Logic, vol.39 (1974), pp.469-477. [23] C. Wood, The model theory of differential fields revisited. Israel Journal of Mathematics, vol.25 (1976), pp.331-352. [24] C. Wood, Notes on the stability of separably closed fields. The Journal of Symbolic Logic, vol.44 (1979), pp.412-416.

Index algebraic 7 function 17 group 26,144 strongly- 46,60 Artin-Schrier theorem 9 atomic over 50 Baldwin-Saxel lemma 23 basis 28 Buium-Manm homomorphism 128 canonical base 19,53,148 cell 16 decompostion 16 Cherlin's conjecture 26 Chevalley's theorem 6 closed set D-closed 45 F-closed 144 λ-closed 144 irreducible 6,46 Zariski 6 constant 39 constructive set 6,8 coordinate ring 7 curve selection lemma 14 definable skolem functions 14 definably connected 18 derivation 38 differential -algebraic 41 -closure 49 decomposition theorem 45 -form 72 full differential Galois group 94 Galois group 87 induced- 97,108 invariant- 96,97 local- 97 -ring 38

-specialization 109 -transcendental 41 differential polynomial 39 generic solution of 49 separant of 39 dimension order property (DOP) 83,143 ENI-DOP 123 division lemma 40 elimination of imaginaries 19-21,53,54,143,146 weak elimination of imaginaries 148 Ersov-invariant 136 exchange lemma 7 extension Liouville 90 normal 89 Picard-Vessiot 59 separable 136 strongly normal 90

field algebraically closed ACF 1 differentially closed DCF 46 real closed, RCF, RCOF 9-11 separably closed SCFe, SCF'e, SCF; 136-140 field of definition 55,137,150 finite covering property (FCP) 52 NFCP 52,143 forking 60,143 formally real 9 free 86 function field 7,78 genus of a 78 fundamental system 58 G-primitive 94 generic 24,100 generic solution 49 heir 60

154

Hubert's 17th problem 12 ideal

/(/) 40 differential 39 radical differential 44 separable 142 indecomposible 100 independent 28,81 jet groups 124,125 K-categorical 1 Macintyre theorem 25 minimal set 146 minimal model 70 model completeness 5,48,137 module definable 32 M or ley degree 7 Nash function 18 Nullstellensatz 5 real nullstellensatz 13 differential nullstellensatz 48 o minimal 11 trivial at a point 35 unstable 7 u -stable group 23,24 p-basis 136 p-independent 136 p- monomial 136 Painleve equation 66 Picard-Vessiot extension 59 prime over 49 quantifier elimination 2-4,47,140 rank depth (RH) 61 differential (RD) 41 Morley (RM) 7,61 U- (RU) 61 Ritt-Raudenbush basis theorem 44 Rudin-Keisier order 85

Seidenberg's embedding theorem 102 semialgebraic 12 semialgebraic expansion 29 semi-positive definite 12 separant 39 simpler 39 Stone space 6 strongly minimal theory 5 strongly minimal set locally modular, modular, trivial 28 strongly regular 84 Tarski-Seidenberg theorem 12 type 6 almost orthogonal 81 definable 51 generic 24,100 orthogonal 81,82 p-base 86 p-dimension (dim(p] K)) 86 a formula represented in a 60 stationary 81 uniform bounding 51 variety 26,137,144 abstract 26 dimension of a 8 morphism of 26 λ-variety 144 simple abelian 119 Whitney's theorem 18 Wronskian 57 Zilber's conjecture 29

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