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i that now c^XD* by (b), while tf>2 >0>8 by (a). Suppose <*> It j < goes over into a system of */7t q for any q. than Then there exists a subscript r higher O has the same rank as
>
Now
by
(a).
let 0*!
> 0>
2
by
while
(a),
0> 2
the smallest integer for which Aj is
4/
higher than Cj and Ai
Thus
*
>
We
by
shall
>Q>3
is
through
(D^^g
8
by
Let j be
(b).
Then
d
for
(a).
and
need the following
fact:
by
higher than Bj. of the same rank as
(b)
if
Finally, 0>i
Q>1
>
is
0> 2
>a>8
by
then
(b),
(b).
Let
(3)
aw
l9 0> 2
,
-..,
Oq
... ,
of ascending sets such that Q>q +i is not Then there exists a subscript r higher than
infinite sequence
.
By are
the
all
of
lemma of 2, the first forms of the 0> q (A in (1)) the same rank for q large. This accounts for the t
with q large has only one form. We may thus limit ourselves to the case in which 0> q with q large case in which
g
has at least two forms.
be of the same rank.
The second forms
Continuing,
more than n forms, that the
q
we
find,
will
eventually
since no
with q large
all
q
has
have the
ALGEBRAIC DIFFERENTIAL EQUATIONS
6
same number
of forms, corresponding forms being of the same This proves the lemma. immediate consequence of this result is that every finite
rank.
An
or infinite aggregate of ascending sets contains an ascending set whose rank is not higher than that of any other ascending set in the aggregate.
BASIC SETS
Let 2 be any finite or infinite system of forms, not all There exist ascending sets in 2; for instance, every non-zero form of 2 is an ascending set. Among all ascending sets in 2, there are, by the final remark of 3, certain ones 4.
zero.
which have a
least rank.
Any
such ascending set will be
called a basic set of 2.
The following method
for
constructing a basic set of
2
can actually be carried out when 2 is finite. Of the nonzero forms in 2, let AI be one of least rank. If AI is of Let AI be of class class zero, it is a basic set for 2. greater than zero.
If
2
contains no non-zero forms reduced
with respect to AI, then A l is a basic set. Suppose that such reduced forms exist; they are all of higher class than AI.
Let
be one of them of least rank.
-4 2
If
2
forms reduced with respect to AI and A2 a basic set. If such reduced forms exist,
them is
of least rank.
Continuing,
we
has no non-zero
A l9 A
then
,
is
2
A$ be one of a set (1) which
let
arrive at
a basic set for 2.
orm
F will
be said to be reduced with respect to the ascending i if is reduced with respect to every Ai, 1,
set (1)
If
Ai
9
in (1); is
f
c ^ ass greater
than zero, a
f
F
Let zero,
2 is
Then
F
an ascending set than AS, else A i9
r.
Suppose that such a form, F, would be must be higher than AI, else must be higher lower than (1). Similarly,
F
to (1).
F
F
F would
be an ascending set lower than
higher than A r set than (1). lower ascending
Finally,
,
be a system for which (1), with AI not of class a basic set. Then no non-zero form of 2 can be
reduced with respect exists.
-
is
.
Then AI,
-
,
Ar F ,
is
(1).
an
This proves our statement.
IRREDUCIBLE SYSTEMS
I.
Let
We
be as above.
-5"
reduced with respect
to
see is
(1),
of
that if a non-zero form, to
adjoined
than
the resulting system are lower
7
the basic sets
2,
(1).
Throughout our work, large Greek letters not used as symbols of summation will denote systems of forms. REDUCTION 5.
we
In this section,
A
deal with an ascending set (1) with
of class greater than 0.
O
form
If a
shall call the
of class
is
form
and
m
of order
in
of
power
ypm
in
G
yp
The
the separant of 0.
dG/dypm
of the highest
ficient
p>0,
,
we
coef-
be called the
will
initial of (?.*
The separant and In
let Si
(1),
i=l,
of At,
We
G
initial of
are both lower than G.
be respectively the separant and
It
initial
r.
,
prove the following result. any form. There exist non-negative integers such that ivhen a suitable linear com1, -, r,
shall
G
Let Si) ti,
and
=
i
be
bination of the Ai and of a certain number of their derivatives, with forms for coefficients, is subtracted from ft
3*
.
.
.
7 f l ...
T*r
r
1
Q
*
reduced with respect to (1). R, limit ourselves to the case in which
the remainder,
We may
S Sr r
1
is
reduced with respect to (1). Let Ai be of class pi, and of order m in yPi Let j be the greatest value of i such that G -
t
with respect to Ai.
We
Let
G
is
be of order h in ypj
is
not
= l,-,r.
i
,
G
not reduced .
=h
the mj, then It will be derivative of Aj, will be of order h in y pj linear in y h, with Sj for coefficient of yp Using the al-
suppose
first
that
h>mj.
If &i
/fith
Pi
gorithm of division, *
for
A^,
.
Later
we
unknowns.
shall If
the
we
find a
^. non-negative integer v such that
have occasion to use other symbols than y^ yn unknowns in a problem are given listed in the order ,
,
w, then w will play the role of yp above, in the definitions of separant and initial for a form effectively involving w.
u, v,
,
,
2
ALGEBRAIC DIFFERENTIAL EQUATIONS
8
=
Sp O
A
where
C AfJ + D (
l
of order less than h in
is
we
a unique procedure,
take
v\
ypj
l
In order to have
.
as small as possible. that pj
Suppose, for the moment, We shall show that integer with Pj
A
is
an
not of
We may limit ourselves to the higher rank than O in y a Also since Sj is free of y a we case in which A4~0. need treat only the case in which y a is actually present .
,
Let
G.
in in
ya yag
in
G
be of order g in ya
cannot exceed g.
If
A
Then the order
.
were
yp/i
G
the same degree Ci would have to involve yag and Ci Af l> would contain terms involving yag and which could be balanced neither by nor by S] G. ,
A
This proves our statement. If DI is of order greater than mj in yp
with
A
in
A
as
of
of greater degree than
A
of
rank than
lower order than
A
(or
in
6)
any
ya
A
in
with
,
1
we
find a relation
ypj and not of higher
a>pj.
For uniqueness,
we
take v 2 as small as possible. Continuing, we eventually reach a
than MJ in yp
,
such that,
J5, t ,
of order not greater
if
we have
^G =
(4)
Furthermore, If
Du
is
if
a>pj,
Du
is
not of higher rank than G in ya u is reduced with MJ in yp
of order less than
*
,
D
respect to A/ (as well as any Ai with i>j). order w/ in we find, with the algorithm
y^,
If
A*
of
a relation
1 with
^ reduced with
Du
respect to Aj, as well as -4/+i, as small as possible. tj
For uniqueness, we take
is
of
division,
IRREDUCIBLE SYSTEMS
I.
We
K
treat as O was treated. For some ?<,/, there such that t t Hffl^K exceeds, by a linear combination of Ai and its derivatives, a form L which is reduced with
are
s
now
,
t
respect to AI, AI^I,
L
exceeds
Ar
,
Then
.
by a linear combination
of
Aj
AI,
and
their
derivatives.
Continuing,
we reach
R
a form
as described in the state-
ment of the lemma. Our procedure determines a unique R. We call the remainder of G with respect to the ascending set
this
R
(1).
SOLUTIONS AND MANIFOLDS
Z
The represent any finite or infinite system. need not all be distinct from one another.* When the forms of 2 are equated to zero, we obtain a system of differential equations, which we shall represent 6.
Let
forms in
2
0. symbolically by 2 In studying the totality of solutions of 2 0, it will be of fundamental importance to have a sharp definition of
=
Let y
solution.
-
-
l ,
,
y
tl
be functions, analytic throughout an in 51, which render each
whose points are
open region 93, form of 2 zero when substituted into the form. The entity composed of 33 and of yi, y n will be called a solution of 0. Thus two systems y^, y n which are identical ,
=
,
from the point
of
different solutions
open region.
view if
For
analytic continuation, will they are not associated with the of
instance,
if
we
take an open region
and use, throughout 93i, we get a second solution of 2
interior to 33,
for 33, *
give
same
t/i,
=
,
j/n
33 t
,
as defined
O.t
What we
mark being
are really considering then, is a system of marks, each associated with a form. Two distinct marks may be associated
with identical forms.
t In Chapter VII, we
shall,
at
one point,
if
they yield
adopt a different definition,
n formal power series, convergent or divergent, a solution, zero when substituted formally into the forms of -T. Many
calling any set of
of our results hold for this definition.
ALGEBRAIC DIFFERENTIAL EQUATIONS
10
By a The
2
of
of 2, we
solution
shall
totality of solutions of
(or of 2 2 and 22
=
mean a
2
will
solution of
=
2
0).
are systems such that every solution of a solution of 29 we shall say that 29 holds 2\.*
If is
0.
be called the manifold
t
2
t
,
COMPLETENESS OF INFINITE SYSTEMS
we prove
the following lemma: , Every infinite system of forms in ylf yn has a finite subsystem whose manifold is identical with that of
In
7.
7
10,
LEMMA.
the infinite system.^
An infinite system of forms whose' manifold is identical with that of one of its finite subsystems will be called comInfinite systems which are not complete will be called plete.% we assume the existence a and force contradiction. complete systems, In what follows,
incomplete.
8.
The system obtained by adjoining forms
2
+
Gi-\ by 2 We prove the following lemma: LEMMA: Let 2 be an incomplete system.
a system
will be denoted
such that, by multiplying each
non-negative powers of FI,
which
Let
F
2+Fi
s
its finite
F
s
,
Let
F
Gm
to
F
be
.
:
,
,
s
by some product of
A
a system
F
be complete, and
,
\-G m
is
8
let
is
obtained
incomplete. it
hold and be
subset
F ~.F
(5)
,
2
Then 2-\-FiF2
is complete.
held by
form
in
6?i,
of in-
t
9,
Hi,-",JSi.
The presence of JF\ F8 in (5) is legitimate, f or if 2 + FI F8 has the same manifold as a system r, it has the same manifold -
r+F
as
l
...
F
s
.
Let
K
(6) *
19
If Si
K
-,
v
has no solutions, every system will be said to hold Si 124 for a comparison of this lemma with a result of Tresse.
t See t If some
.
finite
subsystem has no solutions, the system will be considered
complete.
The product
of
for different forms of
powers of FI, S.
,
F, may,
of
course, be different
2
H
IRREDUCIBLE SYSTEMS
I.
A
which they yield, after the above described multiplications, form a system
be forms
in
such that the forms of
JZi, .--,
(7)
Let L,
and
Consequently
holds
0>
certain
t
.
Now some
in 2, not hold (7).
Ff' ... holds
H
FS*L Then F^
(7).
solutions
of
lutions of (7) are not solutions of
FI
-
F L holds Fs which are
L. Thus
S
L
(7).
so-
does not hold
This proves the lemma. 9. We prove the following lemma:
(5).
LEMMA. Let 2 and 2-\-
F
-
F
3
Then
both be incomplete.
at least one of the systems 2-{-Fi, may evidently limit ourselves to -
,
We
2-\-F
s is
the case
incomplete.
=
of s
2.
and 2 + F2 both be complete. Let if i = 1, 2, be a finite subset of 2 such that 2 + Fi holds + .Fi. Then 2 + jft holds (0>i + 0>i) + *i and 2 + F2 holds Now every solution of (0>! + <&*) + -Fi F2 (0>i + 0> 8 ) + F2 .*
Let 2 + Ft
<J>
+F
+
a solution of (
is
((Pi
+ + + F! F
2)
2
+
+F
.
+F F
,
This proves the lemma.
.
10. Let us consider the totality of incomplete systems of forms in y l9 -, y n 3, According to the final remark of there is one of them, 2, whose basic sets ( 4) are not higher than those of any other incomplete system. Let (1) .
Then AI involves unknowns, else AI would have no solutions, and 2 would be complete. For every form of 2 not in (1), let a remainder with re5. Let A be the system comspect to (1) be found as in of the forms and of the products of the forms of posed (1) be a basic set of 2.
* (
-f (P,) consists of the forms present either in
<J>i
or in <0 S
.
ALGEBRAIC DIFFERENTIAL EQUATIONS
12
2
f
not in (1) by the products Si 2rr used in their reLet n be the system composed of (1) and of the duction. remainders of the forms of 2 not in (I). of
1
Now -Q must be complete. If not, it would certainly have non-zero forms not in (1). Since such forms would be reduced with respect to of
&
then
This means that
4).
(
(1),
hence basic
lower than
sets,
could not be a basic set
(1)
& (1)
would have ascending sets, and 2 would not be an in-
complete system with lowest basic sets. If is a form of not in (1), and
H
form
A
in
then
&,
common with
(1).
H
and E have This means that
A
manifold and also that
The lemmas
2 + Si every
is i,
of
8,
incomplete or Si
and
Ii
is
R
the corresponding
same solutions in and -Q have the same
the
A
complete.
show us now that either some some 2 + It is incomplete. But, for 9
are distinct from zero, and reduced with
+
Si and Then, by 4, the basic sets of 2 of 2-\-It are of lower rank than (1). This proves the fundamental lemma stated in 7.
respect to (1).
NON-EXISTENCE OF A HILBERT THEOREM
One might
conjecture, on the basis of Hilbert's theorem relative to the existence of finite bases for infinite systems ii.
that, in every infinite system 2, there is a such that every form of 2 is a linear combisystem nation of the forms of the finite system, and their derivatives,
of polynomials,*
finite
with
forms
for
coefficients.
We
shall
show that
this
is
not so.
We
consider forms in a single
note in
unknown
y.
(See
first
foot-
2.)
Consider the system
We
shall
n>l
show that no form
linearly expressible in
of this system with terms of the forms which precede
and their derivatives. *
van der Waerden, Moderne Algebra,
vol. 2, p. 23.
is it,
IRREDUCIBLE SYSTEMS
I.
We
13
notice that all of the forms, and all of their derivatives,
are homogeneous polynomials of the second degree in the yt. Also, if the weight of yiyj is defined as i+j'j the pth derivative of yiyj will be isobaric,
Now
with
its
terms of weight
i
-\-j-\- p.
if
the terms in the At, JBi, etc., which are not independent of the yi may be cast out, for they produce terms of degree greater than 2. Again, considering the weights of the various
we
forms,
find that
=
yy*+i
(8)
(pn-2 -----1" C^-i 2n-2 (yi y*) H
Ci
with Ci which are independent of the yj. derivative of y\y* contains a term y\yi n
ft*
-2
= 0.
Continuing, proves our statement.
we
find
every
,
Now ,
the (2n 2)d and none of the
We
other derivatives in (8) yields such a term. that Ci
(yn-iyw )
conclude
be zero.
Ci to
This
IRREDUCIBLE SYSTEMS 12.
A
system 2 will be said to be reducible O and such that neither O nor
that
GH
if
there exist
H holds 2 but reducible will be are not which Systems The system of equations 2 = 0, and also
H
two forms,
holds 2.
called irreducible.
the manifold of 2, will be said to be reducible or irreducible according as 2 is reducible or irreducible.
Example and
Let
1.
2
in the
9
unknown
y, consist oiy\
Let (See Introduction, p. v.) Let Gj and J3i be the remainders for O and i/2
2.
with respect to of order 1 and with a constant be of order to y\
4y.
1.
2.
j/2
GiH
L
is
Then
holds 2.
6?i
y
hold 2.
H respectively be at most a) every y = (#
and HI
Then as
OH
a solution of 2, GiHi,
will
8
if
not zero, must
K
Let be the remainder of O^ HI with respect One can prove now without difficulty that
vanishes identically.
K
Then GiHi
is
algebraically
divisible
ALGEBRAIC DIFFERENTIAL EQUATIONS
14
4y is algebraically irreducible,* one of 4y. As y\ This means, since the 4?/. GI, HI must be divisible by y\ 2 are both unity, that one of initial and separant of y%
by
y\
G,
H
a linear combination of the two forms of
is
Then 2
derivatives. 2.
Example be
We
= 0. u\
uy
irreducible in every
is
use two unknowns, u and y. Differentiating,
we
2ieiW 2
Uiy -\~uyi
2
and their
field.
Let
=
0.
= ^
Certainly
since
it
does not hold 5.
vanishes only for
?/
=
2
= 0,
0.
Neither does
0, if
in the field of rational constants.
fact that u y
=
find
Multiplying the last equation through by y and using
we have
2
u
= 0. We
Thus call
u\ is algebraically irreducible,
2 is
reducible
attention to the
and
of order
in y.
THE FUNDAMENTAL THEOREM 13-
A
systems
2
system
2
i
,
,
2 2S
will if
be said to be equivalent to the set of and every solution holds every
^
2
a solution of some
2it
Thus, two systems with the same manifold are equivalent to each other. We prove the following fundamental theorem. of
is
THEOREM.
Every system of forms
is
equivalent to a finite
irreducible systems.
of Let the theorem be false for some 2. Then 2 is reducible. Let GI and 6?2 be such that G^G*, but neither GI nor G 2 ,
Then 2
holds 2.
is
equivalent to the set
(9)
Thus
at least one of the systems (9) is reducible.
A
re-
ducible system in (9) will be called a system of the first class. *
That
is,
irreducible as a polynomial in y\,y.
IRREDUCIBLE SYSTEMS
1.
There must be a system like 5,
yields
one
when
of the first class, which,
treated
two reducible systems obtained by
or
The
adjoining two forms to 2.
through two adjunctions, we
Some
15
reducible systems obtained systems of the second class.
call
of the systems of the second class,
when
treated like 2,
must yield reducible systems obtained from ^ by three ad-
We proceed that is, systems of the third class. manner, forming systems of all classes. There must be a system of the first class whose forms are
junctions, in this
Let all classes higher than the first. a is either Q or (? be such where HI 2 2+fli, system of first the class the class. second which One of the systems of contains the forms of 2 HI must have its forms contained in systems of all classes higher than the second. Let 2+-Hi+-H"s be such a system. Let an p be found, in this Then the system */*, composed of way, for every p contained in systems of
,
+
H
.
2, -Hi, is
incomplete.
with
a
For,
finite
if
H
2
,
,
Hp,
*P held
subsystem of
2
and
i
<
<
iq ,
then
would hold (10)
This cannot be, since Hi+i does not hold (10). This proves our theorem. One will notice that the proof involves making
an
infinite
number
of selections.*
UNIQUENESS OF DECOMPOSITION 14.
Let a system
2
be equivalent to the set of irreducible
systems
2i,---,2,.
(11)
We
may suppose, suppressing certain of the 2i if necessary, that no 2i holds a Sj with j^i. shall then call each 2i
We
*
See
70.
ALGEBRAIC DIFFERENTIAL EQUATIONS
16
an
and we
essential irreducible system held by 2,
2
a decomposition of
We
shall call (11)
into essential irreducible systems.
shall prove that the decomposition (11)
of 2 into
That
irreducible systems is essentially unique.
is,
essential
,$
if'/2 l?
a second decomposition of 2 into essential irreducible systems, then t s and every &t is equivalent to some 2j. We shall show that there is some &* which holds 2l9 If would have a form which there were not, then each t is
=
-
would not hold ^\ would hold each possible
if
is
2i
Such forms being selected, their product
.
*,
2
consequently 2, thus
t
This
.
is
irreducible and none of the forms holds
2lu
Now
im-
2
l
.
Then let 1^ hold U similarly, some 2 which must be 2l9 since no 2 with i^l Thus 2 und &i are equivalent. The uniqueness holds 2^.
by
t
must be held
12
t
,
t
is
proved.
EXAMPLES 15.
We
some examples involving one un-
shall consider
known, y, in which, in spite of the fact that the systems decomposed consist of a single form, the results are not uninstructive.
Example
we
Let
1.
find, for
2
=
be y\
any solution of 22/22/3
y
(13)
By
differentiation,
-5",
=
2/1
+ 2^ y 27/22/5 + 62/32/4
2y2
(12)
= 0.
jfc
0,
=0,
a 2/3
=
0.
2y3 and
substituting into the result the expression for y\ found from (12), we find that
Multiplying (13) by
2/2(42/3^5
Thus 2
is
12yJ
+ 8y
4
1)
=
0-
equivalent to the set of two systems
y\y, As the only every
field.
solution of
We
shall
2
see
l
is
in
y
0,
^
is
irreducible in
the next chapter that
the
IRREDUCIBLE SYSTEMS
I.
manifold of 2$, which is
irreducible
in
the
is
every
"general solution"
We
field.
note that 2*
hold 2^.
Example
2.
2
Let
==
be y\y*
with a single differentiation, that ~
systems
17
=
y
is
0.
of y\
?/,
does not
We
find,
equivalent to the
two
:
yy*y,
-
?/i
.
2/3
We
call 2i and Z, are irreducible in any field (as above). attention to the fact that the form in - is linear in y$. Example 3. The form y\(yi y) decomposes into the and essential irreducible These two
T/ systems y y. have the solution in common. systems y The above examples might lead one to conjecture, that any 2 can be decomposed into irreducible systems by means of differentiation and elimination. We shall see in Chapter VII
=
that this
is
L
l
actually so.
RELATIVE REDUGIBILITY
A be any system of forms. A system 2 will be be reducible relatively to A if there exist forms G in such that GH, but neither O nor H, holds
Let
1 6.
said
to
and
H
2.
Otherwise
to
A. For
A
instance,
2 if
be irreducible relatively
will
be
2
the form (dy/dx)*
is
said
to
cible in the field of rational constants
if
A
4y, is
2
redu-
is
the set of
all
forms in y of orders 0, 1, 2, but is irreducible in any field if is the set of all forms of orders 0, 1. (See example 1,
A
12.)
We a
see,
finite
The
as
in
13,
that
every system
is
equivalent to
systems irreducible relatively to A. decomposition into relatively irreducible systems need
number
not be unique.
of
For instance
if
A
is
the form
1,
the system
above example, which is relatively irreducible, valent to the two relatively irreducible systems y\ 2. and y\ 4y, y2
in the
is
equi-
4y, yl
ALGEBRAIC DIFFERENTIAL EQUATIONS
18
2
2
can be resolved into relatively is such If irreducible systems whose forms belong to A. that the product of two forms of belongs to A, such a If
consists of forms in^/,
A
A
14. decomposition is essentially unique in the sense of Wherever the contrary is not stated, we shall deal with
irreducibility as defined in all
That
12.
is
A
will
consist of
forms with coefficients in &.
ADJUNCTION OF NEW UNKNOWNS 17. One might ask how the theory of a system 2 unknowns y l9 yn is affected if new unknowns vi,
in the ,
,
n
are introduced, and -5" is regarded as a system of forms in For instance, will the decomposition (11) of 2 the yi, vi. into irreducible systems, when the yi are the unknowns,
continue to be such a decomposition
when
the yi and Vi? To show that the answer to this
question
we
consider an irreducible system prove that it remains irreducible
the
We
yij Vi.
2
the
unknowns are is
affirmative,
of forms in the yi
when
and
unknowns are
the
considered as a system in the
represent 2,
2'.
v^ by Suppose that
yij
G
and
H
are forms in the yi, neither holds 2', but that Let holds 2'.
Vi
OH
arranged as polynomials in the are forms in the yi.
We
t>i,
,
hold
coefficients
We
2
H will
be
,
yn
of
2, arbitrarily assigned func-
terms of
the
G
and
H
in
which the
can be suppressed and the modified
G
GH
be such that neither holds 2', while does. assume thus that no coefficient in O or holds 2.
H
As 2
is irreducible,
coefficient in
the
H
vt.
then,
Evidently,
and
and
with coefficients which
note that the solutions of 2' are obtained by adjoining,
to every solution yi,
tions
Vij,
such that
Vijj
forms,
in (?i
O
G
or
it
will
H vanishes.
have a solution for which no Then we can certainly replace
and H, by rational constants, so as to get two and HI, in the y^ neither of which holds -2". On
IRREDUCIBLE SYSTEMS
I.
19
we can
construct analytic functions n have any assigned values, at any
the other hand, since for which the vy in
OH OH
holds 2', it is necessary that given point, and since This proves that 2 is irreducible. (rtjffi hold 2. f
FIELDS OF CONSTANTS 18. In later work,
will at times be desirable to
it
contains at least one function which
that
cV
We
establish
is
assume
not a constant.
now
a result which will permit us to make no real loss of generality. with assumption that consists e? be purely of constants. Let Suppose
this
^
that is, the totality the field obtained by adjoining x to r, of rational functions of x with coefficients in &. shall
We
prove that then
2
We
is
if
a system
2
irreducible in
start
of
forms in
r
is
irreducible in
r,
S
L .
by proving that
if
G,
of the type
(14)
holds 2, then each Bi holds 2.
with the Bt forms in
F,
(15)
yi(x),'-> yn(x)
be any solution of 2.
Since the forms in
2
Let
have constant
coefficients,
where
c is
a small constant, will also be a solution of
This means that, for any solution (15),
where
is
any constant, vanishes identically
each Bi must vanish identically in x.
in
x.
Then
This proves
our
statement. *
We
shall not
encumber our discussions with references to the areas
in which the solutions are analytic.
ALGEBRAIC DIFFERENTIAL EQUATIONS
20
H
let G and be forms have to prove that one
Now,
We
evidently limit
by (14) and
ourselves to
in &i
of
the
G, H
GH holds
2.
We may
holds 2.
case in which
H by
with the Ci forms in S
such that
G
is
given
r .
G nor JT holds 2. In G and JFf, holds 2 be suppressed. For the which and ft every BI modified G and //, will still hold 2. Then Suppose that neither
let
GH
Since neither
B m nor
GH cannot hold
^'.
B
holds 2, m Cs cannot hold 2, so that This proves that ~ is irreducible in G"?
^
.
CHAPTER
II
GENERAL SOLUTIONS AND RESOLVENTS GENERAL SOLUTION OF
A DIFFERENTIAL EQUATION
We consider a form A in t/t 19. which is algebraically irreducible in r, that of two forms, each of class greater than coefficients in
We
are
We yt
n,
notion
w>l, we
the
of
general
=
n
1,
definition of the general solution will appear, at
first,
=
i
MI,
Our
=
the
introduce
to
going
write y n
=
class
not the product 0, and each with
is,
r.
A.
solution of
of
yn
,
,
y and,
1,
,
if
write q
q.
depend on the order in which the unknowns happen to be arranged,* at least, on the manner in which y is selected from among the unknowns effectively present in A. But it will turn out, finally, that the object which we define is to
actually independent of such order. 20. Let 8 and / be, respectively, the separant and initial of A .t solution of A for which neither S nor / vanishes
A
will be called a regular solution of
We
make
A.
A
that regular solutions of exist. Let A be of order s in z/. Since SI is of lower degree than A in ys 81 and A, considered as polynomials in the unknowns and their derivatives, are relatively prime. ThenJ shall
plain
,
there
is
a
B^
and which, *
That
is
if s
which,
=
if
s>0,
is
than
of order less
on the manner in which the subscripts
1
,
,
n
JBocher, Algebra,
5.
p.
y
are attributed
to the unknowns.
t See footnote in
s in
0, is free of y, such that
213; Perron, Algebra, vol. 21
1,
p. 204.
ALGEBRAIC DIFFERENTIAL EQUATIONS
22
B
(1)
We
the
=
A
for
their
by
x
designate values of
=
ys
,
values
when
fa].
to
which
unknowns and
their
omitting ys then find a
Let
?, fy]
(1),
be taken
number
such that
A
are replaced
the other symbols in
Then, by
at
and
of numerical values
the .
JjS^O. We can
that
any set
attribute
to
derivatives present in A, so
+
to
symbol
to represent
fa]
choose
may
C(SI)
coefficients of the forms in (1) are analytic,
all
the symbol
one
use
shall
which
=
$/
cannot vanish for
In particular, since /S^O, we see by the implicit function theorem that there exists a function
ys
(2)
= f(x\
MI,
-,
ys-d,
at ? [*/], analytic for the neighborhood of ?[^] and equal to which makes A for the neighborhood of ?fy]. Let functions u q analytic at J, be constructed i, which have for themselves, and for their derivatives present Let (2) be in A, at ?, the corresponding values in [y]. considered as a differential equation for y, and let y, f/5 _i be given, at the values which correspond to them in [//]. Then, by the existence theorem for differential equations, (2) determines y as a function analytic at ?, and the functions HI, -, Uq\ y will constitute a regular solution of A. 21. Let Q and if be such that every regular solution of A
=
,
,
,
,
is
a solution of
solution is
of
A
is
OH. We
shall prove that either every regular
a solution of
a solution of H. Let (?i and HI be,
or every regular solution
respectively, the remainders of
G
of A
and
H
with respect to A. Then, as some Sp I t G exceeds G^ by a linear combination of A and its derivatives,* every regular solution
of
A
which annuls GI annuls G]
similarly for
HI
and H. If,
zero, *
and
then,
we can show
At times we t
that either GI or
our result will be proved. shall,
HI
is
identically
Suppose that neither GI
without explicit statement, use symbols,
above, to represent appropriate non-negative integers.
as
p
GENERAL SOLUTIONS AND RESOLVENTS
II.
23
nor HI vanishes identically. As G l and HI are of lower degree than A sj G^R^IS, as a polynomial, is relatively
my
prime to A.
we have
Hence,
=
B
S^O
with and free of ys As in the discussion of (1), we can build a solution of A for which G H^ IS does not vanish. .
But GiHi, like GH, vanishes for every regular solution of A. This contradiction proves our result. 22. It follows immediately, from all
forms which vanish
A
irreducible.
for to
21, that the system of
A
regular solutions of
all
The
this
is
irreducible
system. belongs manifold composed of the solutions of this system will be called the general solution of A (or of A). We show that every solution of A for which S does not
=
vanish belongs to the general solution. Let be any form which vanishes for
B
all regular solutions. linear a combination of derivatives exceeds, by of order at most s in y. C vanishes for all regular
Then some S Ay a
of
C
l
B
We
solutions of A.
have
IPC
(3)
= DA + E, E
E
reduced with respect to A. Since vanishes for regular solutions of A, E, by the discussion of (1), must vanish identically. Thus, as / cannot be divisible by A, C is so divisible. This means that S 1 holds A, so that with all
B
B
vanishes for every solution of our statement.
As we
shall see
solutions with
$=
later,
of
A
with
8^
0.
the general solution
This proves
may
with
of
all
8^0.
forms which vanish for In
a
decomposition
system A, /Sinto essential irreducible systems, let 22 , those systems which are not held by 2lt Then
2lf
(4) is
contain
0.
Let ^i be the system solutions
A
a decomposition of
A
2*, -.., into
2
all
the
of ,
2
t
be
t
essential irreducible
systems.
ALGEBRAIC DIFFERENTIAL EQUATIONS
24
Thus, the general solution of
A
is
not contained in any
other irreducible manifold of solutions of sition
of
A
A.
In a decompo-
into essential irreducible systems, those irreducible
systems whose manifolds are not the general solution are held
by the separant of A. We shall prove that the general solution of A
is
independent
of the order in ivhich the unknowns in A are taken. Suppose that, m being some unknown other than y effectively
we
order the unknowns so that Ui conies last. arrangement, let the manifold of 2j in (4) be the
present in A,
With
this
general solution of A, and let S' be the separant of A. Then S' holds 2^ while 8 holds Suppose that j ^ 1 22 ..., 2t Thus SS holds A. As was seen in the dis.
,
f
.
,
cussion
of
cannot be, since neither
this
(1),
8
nor S'
is
by A.
This proves our statement. In Chapter VI, we shall secure a characterization of the which belong to the general solutions of A with 8 For the present, we limit ourselves to the statement solution. divisible
=
that any solution of
with
84
A
towards which a sequence of solutions
converges uniformly in some area, belongs to the In short, any form which vanishes for all
1
general solution. solutions with
8^
will vanish for the given solution.
see that, in the examples in 15, the systems 22 In each case, the separant vanishes only are irreducible.
We
for
y
can
now
= Q,
and y
=
gives no
each case, the manifold of
29
is
solution
of
22
.
Thus, in
the general solution.
CLOSED SYSTEMS
A
2
if every form Given any system
33.
system
which holds
2
is
will
be said to be closed
contained in 2.*
if we deal only with closed systems. closed system devoid of solutions is the totality
in the study of manifolds,
The only
of forms with coefficients in
*A
given form
is
r
S
.
supposed here to occur only once in 2.
GENERAL SOLUTIONS AND RESOLVENTS
II.
25
A
system which contains non-zero forms, and possesses solutions, will be called non-trivial.
Let
2
be a non-trivial closed system in yt AI, AS,
(5)
Then AI
be a basic set of 2.
A
-
separant or
initial of
any form
-,
class greater than 0.
of
is
of
Let
yn
-,
Ar
-
solution of any ascending set
,
which does not cause the the set to vanish, will be
called a regular solution of the ascending set. shall prove that every regular solution of (5)
We
a solution
is
of 2.
and It be respectively the separant and initial of At. be any form of 2. Then the remainder of G with This remainder, reduced with respect to (5) is a form of 2. be must zero That is, some S^ I*rr G to 4). ( (5), respect is a linear combination of the Ai and their derivatives. Then
Let Let
Si
G
1
G
vanishes for every regular solution of (5). Q. E. D. Suppose now that 2 is irreducible. As no Si or I, holds 2, the product of the Si and 7* does not hold 2. It follows, that, if
2
is irreducible,
Furthermore, if
2
(5)
for all regular solutions of a form, Si Ir G holds Thus, if
2
(5) belongs to
2
is irreducible,
system for which
has regular solutions.* any form which vanishes
is irreducible,
(5) is
so that
then
a basic
2
is
G
2. For, holds 2.
if
G
is
such
the only closed irreducible
set.
ARBITRARY UNKNOWNS 24. Let 2 be a non-trivial closed system in y lf yn There may be some y, say yj, such that no non-zero form of 2 involves only yj that is, every form in which yj appears If effectively also involves effectively some yu with k^j. there exist such unknowns yj, let us pick one of them, arbitrarily, and call it U L There may be a y, distinct from u v such that no non-zero ,
*
;
.
,
* In
Chapter VI, we determine which solutions of (5) other than the regular ones are solutions of 2.
ALGEBRAIC DIFFERENTIAL EQUATIONS
26 form of such ijj
2 let
involves only u and the new y. us pick one of them, arbitrarily,
Continuing, we find a set MI, non-zero form of -5" involves the
If there
and
exist
call it u*.
u q (q
-,
m
alone and such that given the M*, there is a non-zero form
any unknown */;, not among 2 in yj and the HI alone.
*of
Let the unknowns distinct from the
now by y i9
be represented
We
now
yp
list
,
u^'-tUq]
= n).*
to see, in every case,
what
,
-',yp
yi,
speak generally as
shall
taken in any order,
te$,
(p-\-q the unknownst in the order
(6)
We
-
.
exist. It will be easy changes of language are
if 11*
slight
necessary when there are no m. Of the non-zero forms in 2 involving only y and the w, let A l be one of least rank. There certainly exist forms of 2
+
for of class q 2 which are reduced with respect to y t instance any non-zero form in y* and the iti alone is of this Of such forms, let A 2 be one of least rank. type. ;
we
Continuing,
build a basic set of 2,
A i9 A
(7)
We We
t
,A P
,
.
shall say that Ai introduces yi.
shall call
t/i
,
,
w^ a
set
of arbitrary unknowns.
THE RESOLVENT 25. In this section, we assume that purely of constants. Let 2 be a non-trivial closed system.
be ui,
We (8)
,
uq
\
j/i,
are going to
-,
with the
ypj
show
m
F
Let the unknowns arbitrary unknowns.
the existence in
f*i,
,
does not consist
$
of functions
(*p
It will be seen in 30 that when 2 is irreducible, q does not depend on the particular manner in which the w< may be selected, 2 and 5. t See remarks on notation, *
II.
and
GENERAL SOLUTIONS AND RESOLVENTS of a non-zero form G, free of
the existence
27
the yi, such
that either
There exist no two solutions with the same area of analyticily, and with the same Ui,
(a)
^i>
'
>
w/ MI,
for which is
y'i
G
,
*V uq
(10)
of solutions /ii
(y[
y'i)
y'i
yp
-,
and in which, for some
i,
or
,
and for each
exist,
+
+
pair,
y'p)
t*p (y'p
not zero*
is
consider the system of replacing each yi by a new
composed of the forms described, and also the form
t
We
take the
the forms in the
of 2,
That
which the A are unknowns.
z*.
2 by
from
forms obtained
unknown
&
system
in
ti -
?/i,
\
We
just
y'pj
>
//
does not vanish
not identical with
(b) such pairs
'
2/i>
&
is,
zi
involves 3p-\-q
unknowns, namely the Ui, yi, Zi, it. Let be any closed essential irreducible system which 12 holds. Zi, i Suppose that one of the forms yi 1, -,p,
A
does not hold
^.
We
shall prove that
contains a non-
unknowns other than
zero form which involves no
and
A
the
m
A*-
If
A
m
contains a form in the
A
Suppose that has Since
A
alone,
we have
contains no such form.
our result.
=
A m
forms in 2, 1, -,p, has, for j Let Bj be taken alone. a non-zero form Bj in yj and the so as to be of as low a rank as possible in yj. Then Sj, the separant of J5/, is not in A. 1, Similarly let C/, j p, be a non-zero form of all
=
in gj
and the
m
alone,
of as low a rank as possible in
of Cj
is
*
no
Ui exist, this is to
If
A
,
follow the Ui in C/, not in A.
Letting
-
gj
mean
(10) does not vanish for the pair.
that
we if
We
gj.
see that the separant 8j
2 has
take 6r
a pair of distinct solutions,
=
1 in this case.
ALGEBRAIC DIFFERENTIAL EQUATIONS
28
To
fix
our ideas, suppose that yi
sider any solution of
(which is not we have
From
(11)
A i;
(12)
which
in
in
A
we
find,
=
g;(A 9
the
not in A.
Con-
For such a
solution,
derivative of A u
an expression
yn
-,
., Aj,;
,
vanish.
rational in the A t-,
QJ is
-, y,,;
t/ t-,
^
zl7
zp ),
and their derivatives, in each QJ is a power
The denominator
^i-
2/1
Let each Si
^'th
is
which
A} does not
with coefficients in &. of
for
z
*
n
be of order
in y<
and each
d
be
of order
in *f. If
a
involves a derivative of yi of order higher than n, get rid of that derivative by using its expression in
QJ
we can
=
0. the derivatives of yi of order n or less, found from Bi not be of order each so as to transform we Similarly, QJ
exceeding
in
Si
z*,
i
=
The new expression will y\
1,
-,
p.
which
for each QJ,
will involve the
m,
have a denominator which is a product of powers of Let # be the maximum of the *i; Si, Sj, i 1, ,!>.
integers n,
Let
Si.
h
Let k be the
total
number
of letters y
Zij
which appear in
the relations (12), transformed as indicated. Then h>k. consider the first h of the relations (12).* (That is,
We
= 0,
we
let j
of
powers of yt
1,
for the second
h
1).
*u the Si,
members
Let D, an appropriate product Si', be a common denominator
of these relations.
*=f
<*> *When
-,
/
= 0,
(12) is (11).
'
We
write
GENERAL SOLUTIONS AND RESOLVENTS
II.
j
=
o,
,
&
nomials in the
and the Ej, be written as polywith coefficients hp and the w,. Let m be the
letters y^, zy present in them,
which are forms
maximum
D
Let
1. A:
29
A2 ,
in
,
of the degrees of these polynomials (total degrees
in the y#, zy).
Let a represent a positive integer to be fixed later. The number of distinct power products of degree ma or less, in k
letters, is*
(ma + k).~(ma+l)
(U)
^
Using of the
us form expressions for
(13), let
Ay
in (13) of degree
all
power products
Let each expression
or less.
be written in the form (15)
Jr-
Then F, as a polynomial at most
in the y^, zy,
will
be of degree
ma.
The number degree a or
of
h
of the
power products
letters
A v-, of
less, is
fia\ (16)
\
a
*
*/
i
*
a
*
\
i
I/ .
Now
(14) is a polynomial of degree and as As of degree h in
h>k
.
will exceed (14)
is
if
large.
k in
,
whereas (16)
is
are fixed, (16)
w, A, Let a be taken large enough A:
for this to be realized. If
now
the
F in
(15) are considered as linear expressions
we
will have more Hence the linear power products. That some linear are linearly dependent. is, expressions combination of the F, with coefficients which are forms in
in the
power products
in
the yy, zy,
linear expressions than
F
lp and the ut, not all zero, vanishes identically. linear combination of the power products of the
Aj
The same
will vanish for the
Jiy
written. *
Now,
this
Perron, Algebra, yoL
solution
last
1, p.
of
A
for
which (11) was a form
linear combination is 46.
H
ALGEBRAIC DIFFERENTIAL EQUATIONS
30
m
in the
^
and
which
products of the
1*\j
is
not identically zero, since the power are distinct from one another.
in
H
is
in
Thus
A, so Let AI
is in
H
that
,
Ar
,
be
a
A.
This proves our statement. closed
into
of
decomposition
A each not conAr each contain tain some form yi Zi and let ^5+1, A i = 1, in Let be a form non-zero Hi every y s, H K Let the and m Hi Aj. involving only Let A\
essential irreducible systems.
,
s
,
,
.
Zi.
t
We then
in
K be ^
the
in
K.
that,
does not vanish identically in the H{. written as a polynomial in the t^-, with forms
JST
Let
.,
s
.
show the existence in S of p functions when each A$ is replaced by ^ in K,
wish to
% Mp such
Mi,
.
iy
-
as
we
If
coefficients.
can
fix
each
Let
^
L
in
?
be one of the coefficients so that
L
does not vanish,
our result will be established.
be any non-constant function in $, and let a be 51 at which is analytic and has a non-vanishing derivative. Given a sufficiently small circle with a as center,
Let
f
a point of
can be expressed as with constant coefficients. Then
any function y, analytic a power series in
in the circle,
be approximated uniformly within the in
Thus
.
ko,
km,
,
m
if i
a
is
1,
-,
p
vanish identically in the
circle
by a polynomial and if
large are arbitrary constants,
integer,
sufficiently
ty
if
each
L
cannot
replaced in
A* is
L
by
L would vanish if the ^ are any functions analytic above circle. Thus there must be integral values of the tij for which L does not vanish. Every polynomial in with integral coefficients is in This shows the existence Otherwise
in the
r
.
of the required /^.
The
solutions of
solutions A/
= pj
of the of ^i,
-
= =
for A/
Ai -
,
^;,
j
=
1,
-
,
p, will be the
for */ Now, the solutions with f*/. v/5 have u* which cause to vanish the
GENERAL SOLUTIONS AND RESOLVENTS
II.
=
31
in K. The solutions obtained by putting A; /*; 1 of r even with A/ 8 +i, ^, i p. ^, have yi When every Aj contains every yt Zi, we take (7=1,
form
4
=
,
A
=
,
=
,
,
=
We
0. Pp have thus the result stated
26.
We
/*i
=
at the
head of
this section.*
now
relinquish the condition that a non-constant function shall
m
Let us assume that existence
We
exist.
of forms 0, M\, such that, for two distinct -
Mp
-
,
Mi(y[-y'l')+
-
are going to prove the
+Mp
-
contain
in the in alone, with
,
solutions (9) for which
vanish, (17)
&
O
G^Q,
does not
y'A
(y'p
not zero.
is
The
the form
Mi, by Mi Let
25 holds through the construction of are going to prove the existence of forms
discussion of
-
,
We
K.
Mp
m
alone, such that, when A, is replaced the K, resulting form 6 is not identically zero. be arranged as a polynomial in the A^-, with forms
in
K
in the
in the ui, A 2
,
,
hp as coefficients.
Let U\H be the highest
derivative of u^ which appears in any of the coefficients. Let k be an integer greater than h. Then, if AI is replaced in the in and A 2 becomes a form ..., ip by MM, which is not identically zero. Similarly, if we replace A2 in
K
KI by a
K
sufficiently high derivative of MI
zero form
K
and A 8
in the t^
2
in succession
by
tain a non-zero form
O.
Continuing as in
25,
which which
A/
=
Mj, j
satisfy A;
of -^,
-,
A
9
=
=
have
1,
Mj.
m
,
,
Ap
.
,
we
obtain a non-
Replacing^,
high derivatives of HI,
sufficiently
we
p
,
we
^> ob-
n
in
are the solutions of the
Ai
see that the solutions
-,
,
,
of
=
Now
the solutions with A; which cause (7 to vanish. The
Ji}
so-
The following example shows that 2 may have many solutions with given Ui, and that a G may exist, such that, for #4=0, there is only conone solution for given t*<. Let the unknowns he Wi, ws y\. Let *
2
,
sist
of all
forms which hold MI y\
U*
.
Let
= =
G = HI.
Then w t
,
w
is
a set of arbitrary unknowns. If Wi Wj 0, y\ may he taken arbitrarily, but, for given M I? us with (74=0, there is only one y\*
ALGEBRAIC DIFFERENTIAL EQUATIONS
32
i
=
=
Zi, ^rj even with A/ Mj have yi our statement. This proves p 27. The results of 25, 26 permit us to state that if
lutions of ^5+1,
=
1
-
-
,
-
,
9
.
,
either (a) o?
does not consist purely of constants or
(b) there exist
w,-,
P
then triads of forms G, P, Q, exist with O and not in 2, and # free of the y, such that, for any two distinct solutions of 2, with the same w, such that neither G nor
P
vanishes, the expression
of x.
For
instance,
Q/P
if (a)
yields
holds,
two
we can
distinct functions
take
P=
generality of our
It will essentially increase the
The following
1
and
work
to
a non-trivial example in which Let r be the tois of class greater than 0. Let the unknowns be y rational functions of x. of tality
use general forms P.
is
P
9
y2 and let 2 consist The solutions are y l
but arbitrary.
We
of
=
all c,
forms which hold yu und y%. d, with c and d constant y*
take
=
=
If
1.
the expression Q/P gives distinct functions of 0. solutions of 2 with
x
for distinct
P^
In certain cases in which
and
in
which no
m
&
consists
exist, there
may
purely of constants no pair P, Q as
exist
For instance, let & be the totality of conLet the unknowns and 2 be as in the preceding stants. for every solution. example. The yy are all zero for j We therefore lose no generality in seeking a P and Q of order zero in yl9 y*. For any such P and Q, Q/P will yield the same result, for infinitely many distinct pairs of condescribed above.
>
stants
t/i,
t/2
.
In developing the theory of an irreducible system 2 for the case in which & has only constants and there are no Ui y
two courses are open
to us.
If
we
adjoin
x
to
r,
then,
II.
by
2
18,
Working
GENERAL SOLUTIONS AND RESOLVENTS remain
will
enlarged
field.
we can secure a P and Q. we can introduce a new unknown MI and 2
in the enlarged
Again, by
the
in
irreducible
33
17,
field,
remain an irreducible system. After either type of adjunction, the theory which follows will apply. 28. In 28, 29 we deal with a non-trivial closed irrewill
We
ducible system 2. assume that either does not consist entirely of constants, or r
(a)
(b) arbitrary
unknowns
exist.
We We
take a triad 6, P, Q, as in 27. introduce a new unknown, w, and consider the system obtained by adjoining the form Ptu be Q to 2. Let
A
the system of for
forms in w, the ui and y{ which vanish We shall prove that with P^O.*
all
of
solutions
all
A
12 is irreducible.
Let B and C be such that BC holds 12. For s appropriate, P B minus a linear combination of Pw Q and its derivatives, We obtain similarly, from a P*C, is a form R free of w. an 8, free of w. Then US vanishes for every solution of 2 S
with
P^O,
since every such solution yields a solution of or 8 is in 2. so that either
R
is
PES holds 2, in 2, P B holds
of
A
with
Hence
R
8
We
P^O,
notice
Hence
12.
so that
B
B vanishes for
is in
that those forms
12.
of
12
all
12.
If
solutions
Thus 12 is irreducible. which are free of w
are precisely the forms of 2. In particular, non-zero form in the alone.
12
contains no
m
We in
w
are going to show alone. and the
\
1, ,
u
-, ,
contains a non-zero form
PJ be a non-zero form of
of
minimum rank
separant of Bi. Consider any solution of
*
12
m
Let Bi, i only y{ uv
vanish.
that
For such a
Of course, forms in
&
may
which we have
12 for
solution,
in
also vanish
yr
PS
when
involving be the i
S
Sp does not
L
P
2
Let
0.
ALGEBRAIC DIFFERENTIAL EQUATIONS
34
For the jth
derivative of w,
(18)
J
we have an
expression
=
=
Using the relations Bi 0, we free each Qj from derivatives of each y i of order higher than the maximum of the orders of Qj P and Bi in y r Each w. will then be expressed as a quotient of two forms, the denominator being a product Sp If we use a sufficient number of powers of P, Si, -
,
the
of
relations
as
(18),
.
we
transformed,
just
have
will
more w, than there are y. in the second members. Using the process of elimination employed in 25, we obtain a nonin w\ Mi,-,w zero form which vanishes for every
K
solution in
tf
&
of
&
and as
,
We now
29.
Sp ^
with PSi is
irreducible,
list
the
As
0.
in
.XT is
unknowns
Sp
P/Si
is
not
.
in the order
in
and take a basic set for Q,
A,
(19)
Here, w,
j/ 1?
remarks If
A
replace
,
of
yp are introduced
in succession.
(See final
24.)
not algebraically irreducible, we can evidently by some one of its irreducible factors. We assume,
is it
therefore, that
We
4,...,^.
A
is
algebraically irreducible.
Ap are of order prove that AI in y l9 yp and, indeed, that A. is of the first degree in yr Thus, since Ai with i>\ will be of lower degree in yj than i y each equation A. with j expresses yi rationally Aj u q and their derivatives. in terms of iv\ Wi, The determination of the manifold of 2 will in this way are going to
-
,
,
-
,
,
=
<
,
be made
to
solution of
depend on the determination of the general which equation will be called a resolvent
A = 0,
of 2.* * If
of A.
A
is
any system equivalent to 2, we also
call
A
=
a resolvent
GENERAL SOLUTIONS AND RESOLVENTS
II.
35
Suppose that A t is of order higher than zero in ylm Consider any regular solution of (19) for which does of the final remarks not vanish. such 23, regular By solutions exist. Let be any point at which the functions
PG
2jf
in this solution
and the
coefficients in
P, 0, A, A,, and for which, if A, Si and It those
are analytic,
and
initial of
POSS, w
Without changing
.
..
Split
m
or the
Ap
..-,
8 and /
are the separant
of Ai,
"-
Ip
+
0.
we can
in the solution,
alter
in A^ other slightly the values at ? of the derivatives of y than the highest, and obtain a second regular solution of 0. That is, we can solve A for (19) with ,
,
=
PG
with the modified
A2
into
yi
for
initial conditions,
A2
solve
,
=
which obtained
2/2
determine
continuing,
with the same in
the
each
first
^
the resulting initial conditions
substitute
regular solution,* and, is nothing but an
This
yt.
application of the implicit function theorem, and of the existence theorem for differential equations. Thus, we would with the same ut, with have two distinct solutions of ,
PG
This contradicts the and with the same w. fundamental property of the triad G, P, Q. 4"-
0,
Hence, in yj
3
^
A
is
of
order zero in
Similarly, every At is
t/i.
is of lower rank Furthermore, than Aj for j
of order
zero in
as At
yi.
*'
We We
shall
now prove
start with
Ap
.
that each Ai
is
Suppose that
Ap
linear in yi. is
not linear in yp
.
P
with respect to (19). Then every regular solution of (19) which causes either of the forms P, PI to vanish, causes the other to vanish.
Let PI be the remainder for
*
That
is,
the highest.
with the same values at
for y^
and
all its
derivatives but
ALGEBRAIC DIFFERENTIAL EQUATIONS
36
we can show
If
that the ascending set
A,
(20)
A 19
,
Ap-i
has a regular solution 1,
for
which
we
shall
Ap has two
W\
Uq]
,
7/i,
-,
distinct solutions in
yp -i
yp with Sp Ip Pi
have forced a contradiction. we cannot get two distinct solutions
If
must be that for every regular solution the equation vanishes.*
Ap
=
of this
0,
type,
it
Ip ^ 0, which Sp PI G
of (20)
has a solution in yp for
^
with
Sp PI G with respect to A p conThen C is of zero order in an ascending set. is of and lower Every degree than A p in yp every yi common solution of Sp PI G and Ap is a solution of C. We Let
C
be a remainder for
sidered as
.
note that C, like
w
Sp
,
PI and
Ap
,
is
not of higher order in
than A.
all forms not in fi, of zero order in the yi, which are lower degree than A p in ypj not of higher order in w than A, and which, for every regular solution of (20) with
Of
of
have an annulling function yp in common with Ap Then be one which has a minimum degree in yp let would must be at least of the first degree in yp else IP vanish for every regular solution of (19) and would be in
IP
^
,
D
D
.
D
,
.
Let
K
be the
would not be
of
priate,
D. Then a minimum degree
initial of
K m Ap
K
is
in
yp
not in .
For
,
m
else
D
appro-
= ED + F,
E of lower degree than Ap in yp and F of lower degree than D in yp Every annulling function yp of Ap and D makes F vanish. Then F must be in
with
.
.
ED
Thus is in
(21) *
.
must be Let
E
in fl,
so that
= H + Hiy Q
E, which
is
not zero,
+--
Because (19) has regular solutions, (20) has regular solutions with
1,4=0.
GENERAL SOLUTIONS AND RESOLVENTS
II.
37
with the Hi forms free of yp and of order in w not greater than that of A. We understand that Ht ^ 0.
Km Ap
As
ED
is
of higher degree in
identical with that of
is
Then Ht
not in
is
yp than F, the initial and hence is not in
of
K m Ap
.
.
Evidently a non-negative integer a t exists such that, when a suitable multiple of A _ l is subtracted from 1 i9
I^_
p
= Q,
H
remainder is of lower degree than Ap-i ap -\; In the same way, we find integers a2 , a such that when a suitable linear combination of the forms
i
-,t, the
in 2/^-1.*
,
of (20) is subtracted
i
=
.
.
.
,
y
t,
from
the remainder
is
reduced with respect to
,
we
As
IMS
..! is in
(20).
contains a form
see that
Hi
reduced with respect to (20) and with Hi not As t is less than the degree of Ap in 0). is which not zero, is reduced with respect to (19). yp EI, This contradiction ( 4), proves that A p is linear in yp with each
(hence not
in
,
.
We now yp -i. of yp
consider ^lp-i, assuming that it is not linear in Since PI is of lower degree in yp than Ap , PI is free
.
It
must be
that, for every regular solution of
A,A 19
(22)
with Ip-i
^ I PiG /Sp-i p
0,
Ap-i =
has a solution in yp-\ for which
The proof continues as
vanishes.
for
Ap
.
In dealing with Ap -i, we consider that both PI and Ip are free of yp -i. The proof continues as above. Thus every Ai is linear in yi and each yi has an repression ,
rational in w\ u i9
,
uq and
cients in &. *
Note that Jp-i
is
free of
yp
.
their
derivatives,
with
coeffi-
ALGEBRAIC DIFFERENTIAL EQUATIONS
38
We
notice that, if
u l7
uq
-,
iv\ */i,
\
a solution of &, then u i9
is
-,
uq
yp
,
\
w
belongs to the general
solution of A.*
For,
if
K
a form
in
w
and the
every regular solution of (19), and so
When Q is
in
is
2,
w
=
rational in the w#. The introduction of
resolvent
every for
Q.
in
is
for
K vanishes
then
a resolvent.
is
the
vanishes
114
solution in the general solution of A,
Then each
accomplishes
yi
the
following:
reduces the study of an irreducible system to the study of the general solution of a single equation. Of course
(a) It
2
for solutions of
responding of some At
w
with
=
P
0,
there
and, for other solutions vanish.
may
We
may
be no
cor-
2, the initial shall gain information as of
to these exceptional solutions in Chapter VI. (b) It leads to a theoretical process for constructing all irre-
ducible systems
(
32).
an analogy between y l9
(c) It creates
,
y p and a system
p algebraic functions of q variables. It is well known, in short, that, given such a system of algebraic functions, we can find a single algebraic function in terms of which, of
and
of the variables, the functions of the
system can be
expressed rationally. furnishes an instrument useful in the solution of forIt (d)
mal problems. INVARIANCE OF THE INTEGER q 30. in
any
We
We
consider a non-trivial closed irreducible system
field
2
r .
propose to show that,
if
arbitrary
unknowns
exist,
the number q of arbitrary unknowns does not depend on the manner in which the in are selected. *
Here we consider
A
as a form in
to
and the
u< alone.
GENERAL SOLUTIONS AND RESOLVENTS
II.
Let
will
It
exist.
Ui
and
ui
prove
m
As
form a resolvent for 2.
given any
that,
yi,
2 which
there exists a non-zero form in
We
to
suffice
unknowns among the
39
Let us consider the regular solutions
involves only the
Zi.
exist, this is possible.
of (19).
Every
zi
in
such a solution has a rational expression in tii, u q w. If a Zi happens to be a H, say M/, the expression for that \
,
Zi
is
simply
(23)
On Zi)
Zi
We
iij.
=
UL
(;;
Qi
write
uq ),
-
,
,
differentiating (23) repeatedly,
which are rational
relation
=
A
we
0,
A
None
we
1,
,
g+1)-
get expressions for the Making use of the
and mj.
in the wj
as not to
transform these relations so
w
contain derivatives of the order of
=
(i
of order higher than
r,
where r
is
w.
in
obtained will have a deno-
of the expressions thus
minator which vanishes for a regular solution of (19).
and only q of the Mi, it enough (and then will become more numerous than the ug
Since there are q~\follows that
we
if
transform), the
and w, MI,
Zij
-
-,
wr
the
Zi
z,-,
*
It follows, as in
in
of the
1
differentiate (23) often
25, that there exists
which vanishes for
all
The form thus obtained belongs is
a non-zero form
regular solutions of (19).
to 2.
The invariance
of q
proved.
The assumption that 2
is
irreducible
is
essential.
For
in-
stance, consider the system
MI
2/i
=
^2
2/2
=
u3
2/2
=
0,
unknowns MI, M 2 M 3 yi, y%. These equations impose no relations either upon the yi or upon the m. Thus MI, f arbitrary unknowns. M2, MS and 2/1, 2/a are two sets
in the
,
;
4
ALGEBRAIC DIFFERENTIAL EQUATIONS
40
ORDER OF THE RESOLVENT 31. for
We work with any non- trivial closed irreducible system 2
which
triads
Considering
2
G, P, Q, and therefore resolvents, m and t/i, let
exist.
as a system in the
^i,---,4,
(24)
be a basic set for 2, the separant and
and Ii respectively. Let the order of Ai
in yi
initial of
Ai being Si
Let
be n.
We shall prove that every resolvent of 2 is We begin by proving that n contains in
of order h in w. a non-zero form in not u order w does exceed h. whose w\ ?(i, -, q for which Consider any solution of
PS,
(25)
For such a
solution,
.
Spli
We
+
0.
we have iv
(26)
Ip
=
|.
propose to show the existence of forms
=
of order not exceeding r< in y<, i 1, for which (25) holds, any solution of
w
(27)
=f
R
and
T
each
such that, for , p, T is not zero and
.
Let Qi and PI be the remainders of Q and P respectively, Let Qi be obtained by subtracting a linear combination of the Ai and their derivatives from
relative to (24).
and
let
P
t
be obtained similarly from
rp
p
p.
GENERAL SOLUTIONS AND RESOLVENTS
II.
Then,
if
(25) holds,
we have
a s? = -^~~ 1
w
(28)
41
i Tp *-.
i
For
H
T
and
in (27)
we take
the numerator and denominator
in (28) respectively.
We
from
derivative of w,
an ex-
If Uj is the remainder of Bj with respect to (24), write (29)
we can
find,
(27),
for the jth
pression (29)
wj
(30)
=
wj
=
%r,
Ip is a product of powers of T, Si be Consider (27) and the first h relations (30). Let a common denominator for the second members in these where Wj
-
-
.
D
h
-\~
We
relations.
1
(3D 3
=
write wj
0,
,
=
JL,
h.
Let D, the Ej and the Ai be written as polynomials Let yij with coefficients which are forms in the ut.
the
be the maximum
derivat"'* of yi
be a positive integer, to be fixed
Let
form
all
m
of the degrees of these polynomials.
For convenience, we represent the nth Let Ai be of degree vt in z\. let us
in
power products
later.
in the wj of degree
by
*<
In (31),
a or
less.
Let the expression for each power product be written
in
the form
-.
(32)
Then each
F
is
a polynomial in the yy, of degree not ex-
ceeding met. Let each expression (32) be written (33) 4*
ALGEBRAIC DIFFERENTIAL EQUATIONS
42
Consider a particular F, and let it be written as a polynomial in zp Suppose that its degree d in gp is not less .
than m.
Ap
Then, as
we
are considering,
=
for the solution of
Q
which we
have, letting
the relation (34)
with the Ji free of zp form
we may
,
write the numerator in (33)
in the
(J Ip
(35)
Since
Ip
of
is
+
...
+ Jd Ip e*)I?>-i.
degree less than
m
in the y
each term
in
of degree less than m(a-\-l). d ~ v* in (35). As d is of Jd l z* replace by d p and as in is of d the not yu degree exceeding d ~ Vp is at then of less than most m, degree degree d into in the yy. over Thus, (33) goes
the parenthesis in (35)
We
is
Mz
J
J
M
ma
J Mz
w(+l)
where FI
is
less
degree less than m, operations,
+
of degree less than w( l) in the y^ and of is not than d in zp If the degree of JP\ in
^
.
we repeat the above we get an expression TT jmnt
After
operation.
t^
"" ^
with Zf of degree less than
w(
+
<)
m
and of degree
in zp
The numerator
in the y
in (36) is of
less
than
degree in
the y$ less than
< 2m
w(4-0 + w(m
A
=
1
2 .
Z> Jj we can write each power Thus, if we let or less, in the form product in the Wj, of degree <37)
,
-
GENERAL SOLUTIONS AND RESOLVENTS
II.
where JTis than
less
We
of degree less
m
now
2w2
than
in the
43
y{j and of degree
in zp .
write each expression (37) in the form
(38)'
Tim a
T)a J
and employ, with respect to Zp-\, the procedure used above. find for each expression (38), an equivalent expression
We
with
D
t
=
J>i
l2
L
and with
^
m
the y and of degree less than {j
we
find
4m
of degree less than in
^
and
z __ r p
8
a in
Continuing,
an expression for each power product of the wj
W (40)
-^r
where
W
degree
less
than
The number less
than
m
in
w of
in
and
#,
i
=
1
p. Let
,
,
in
power products
each
Hence the number or less,
m p+1 a
of degree less than 2 p
is
mp
letter, is
of
and of
y^
represent 2^
,,
,
m p+1
.
of degree
.
power products
of degree less
t
c
in the
m
than
in
ca
of the y.. of degree
each ^,
is
not more
than (ca-\-h) ... (ca (41)
This
+ 1)
hl is
because the
..
ly
with
^'
.
t
are h in number.
On
the
other hand, the number of power products of degree a or less in the h 1 Wj is
+
(42)
/7
,
-,\,
+
As
l in and (41) only of degree h y (42) is of degree h (42) will exceed (41) for a large. This, as we know from
25,
implies the
and the
m
existence
of
a non-zero form of
alone, of order not exceeding
h
in w.
in
w
ALGEBRAIC DIFFERENTIAL EQUATIONS
44
This shows that the order in
iv
of the resolvent
does not exceed h. Suppose that the order For each yv we have an expression
with
d
ceeding j
= 0,
A
is
k
< h.
f
and Di forms in w\ MI, i^, of order not exin w. We obtain from (43) expressions for the y^., n1, which are rational in the wj,mi, with ,
it
,
of the
powers
we
=
yt
(43)
of
=
A
A
as denominators.
depress the orders in
not exceed
w
Using the relation
The transformed expressions
fc.
4 = 0,
of the numerators until they do will
A
have de-
and 8. nominators which are power products of the in the By an elimination we obtain a non-zero form
W
., hence to 2. This W, which of order less than r t in each y,-, is reduced with respect HI
2/i,
which belongs
to -
is
This
to (24).
We
is
impossible.
have thus proved that the order
in
w
of
every resolvent
h.
is
We
now
when
u q are selected, the quantity on the manner in which the f*i depend Tp to the remaining unknowns. are subscripts 1, ,# assigned This follows immediately from what precedes, except that we have to prove that, when no m exist and of has only
+
say
+
that,
UL,
-
,
does not
constants,
n-f
What we
do
is
-\-rp is
remain irreducible, remain a basic set.
The degree
independent of the order of they*. a new unknown, u lt 2 will
introduce
to
iii
be an arbitrary set, and (24) will The methods above then apply. will
of the resolvent in
WH does depend on
P and
Q.
Consider, for instance, the system, irreducible in the field of all
rational functions
As the manifold
is
^
w = yi.
= x + a,
y*
= (x +
2
a)
,
we may
evi-
=
0. 1 The resolvent becomes wl dently take On the other hand, if we take w the resolvent 2/1 2/2 , becomes of the second degree in w\.
= +
GENERAL SOLUTIONS AND RESOLVENTS
II.
The order For instance
of the resolvent depends on the choice of the ?*.
2/112/2
=
irreducible in the field of all constants.
is
we we
45
get a resolvent of the first order. get a resolvent of zero order.
we let HI we let u
If
If
= =y
y%
,
L ,
CONSTRUCTION OF IRREDUCIBLE SYSTEMS
We
32.
shall establish a result
a converse of the result of
which
to
is,
some
A
Let
be an algebraically irreducible form in w, effectively involving w. Let (44)
yi
where the Pi and
:
=
--,
i
are forms in MI,
Q
extent,
29.
,
uq
\
MI,
=
w
,
uq
;
,#,
1,
and where
Q
does not vanish for every solution in the general solution Let MI, ---lUg] w be any solution in the general of A. For this solution, solution of A which does not annul Q.
we
obtain,
Let for all
GH
Let
from
(44), functions j/i,
,
yp
.
be the system of forms in the MJ, y<, w which vanish HI, yi, w. We shall prove that .,is irreducible. hold
If
.
we
substitute (44) into
(?,
we
get
We
M w and Z7 a power of Q. with T a form in MI, V/W. For any lit, w, T V vanishes. have, similarly, Then vanishes for every solution in the general solution ,
H=
ff ;
TVQ
of
or
A. Thus either
V
does.
T
vanishes for every such solution of A, must vanish for all Consequently one of G,
w, y^ w. Hence By the method
&
H
is
of
non-zero forms in the of all such forms
system.
What
is
irreducible.
25, M,-,
we can show
y< alone.
that
contains
The system 2 composed
and the zero form
is
a closed irreducible
more, the theory of resolvents shows that
ALGEBRAIC DIFFERENTIAL EQUATIONS
46
yi, yn can be obtained have thus a theoretical process for con-
every closed irreducible system in in
this
We
way.
,
structing all closed irreducible systems.
IRREDUCIBILITY AND THE OPEN REGION 33-
The question might be raised
whether 2, irrecan be reducible in &
as to
for the open region 21,
ducible in
21
some open region 2li in 21. We shall show that the answer is negative. Without loss of generality, we assume 2 non-trivial and
for
Also, we assume, adjoining x to & if necessary, that does not consist purely of constants. Let a form vanish for all solutions of 2 which are
closed. oF
K
K
We
shall prove that vanishes analytic in a part of 2li. in 2. for all solutions of 2. that is not Suppose
K
We
construct a resolvent, and consider (19).
K
mainder of with respect to A. On the other hand, by
K
29), vanishes for every solution of
lytic in a part
of 2Ii.
not divisible by
Thus
Then
if
P
solutions of
and
2
A Q
2
is
impossible,
are forms such that
analytic in a part is
is
be the re-
not divisible
A
Avhich
is
ana-
because (45)
is
20).
(
for all solutions of 2.
2, so that
This
v
Jfi
Ip
K, 8 II,
(45) (as in
(19).
Let
of 2li,
This means that
irreducible in 2d.
PQ vanishes for all then PQ vanishes either P or Q is in
CHAPTER
III
FIRST APPLICATIONS OF THE GENERAL THEORY
RESULTANTS OF DIFFERENTIAL FORMS 34. In algebra, in developing the theory of resultants of systems of polynomials, it is necessary to deal with polynomials whose coefficients are indeterminates. So, in connection with resultants
of
pairs
of
differential
y
we
forms,
shall find it desirable to deal with general forms. purposes, it will be convenient to define a general
For our in
form
as one of the type
A with n 2> Pi,
i
=
=
OQ
+ MI y + #2 P^ H
where the
1
0,
,
n, (Po
ai
=
\-
an
Pn
,
are indeterminates and where the 1,
PI
=
are distinct
y),
power
products in y and its derivatives. By an indeterminate, we mean a symbol which can be replaced, when it is desired,
by an
arbitrarily assigned analytic function.* Consider a second general form
B We
= + Jb
propose to find a condition upon the a
-
t
and
bi,
ne-
A y and B. Let the ca and bi be considered now as unknowns, and let A and B be considered as forms in the a*, bi and y in any cessary for the existence of a
field
\
We
shall
common
solution in
of
show that the system
*
It is understood that, when the a* are replaced by, analytic functions, the replacing functions have a common domain of analyticity. In defining a general form, we do not use the notion of a coefficient field.
47
ALGEBRAIC DIFFERENTIAL EQUATIONS
48
B
A,
(1)
irreducible.
is
X
Let
=
fl
(2) If
and
i
and
of
members
T
Pn
an
y
in (2), there result
(3)
i,
,
such that Xi YI holds
an
hy in Y
X
&i,
\
=
b
\
We
holds (1).
have
bm
Qm
.
and by the second two forms Xi and FI in
are replaced in
&
XY
be forms such that
,
&w
^
;
But as the unknowns
(1).
may
(3)
b be be taken arbitrarily, as analytic functions, and a B 0, it must be determined by (2) so as to make A Then one of i9 Ft must that Xi YI vanishes identically. ,
= =
X
vanish identically, and one of Z, Y must hold (1). Let 2 be the system of all forms which hold (1).
can be shown, by the method of non-zero forms in the 04 and bi alone. (2),
25, that
it
On
2
the other hand,
(4)
i
;
,
2
Using contains
contains no non-zero form in
,
an
bQ
;
,
,
bm
.
Suppose that such a form, C, exists. Let the unknowns b m not (4) be taken as analytic functions, with &i ,
zero, so that in y.
that
^4
(7^0.
Then
=
?/,
in all
=
has a solution
we can
determine 0$ so
certainly J?
Using any such solution 0. C cannot exist.
,
Suppose now that $ is the field of all rational constants. Let (4) be the arbitrary unknowns, and let
R,
(5)
be
a
basic
set
for
2,
R
U
and
U
introducing
a
and y
respectively.
We
R
is algebraically irreducible and that are relatively prime integers. This determines For, let 8 be any uniquely, except for algebraic sign. other form which satisfies the conditions placed on R. Ther
assume that
its coefficients
R 8
and
R
are of the
same rank.
The remainder
of
S
with
APPLICATIONS
III.
49
B, being in 2, must be zero. Then J being the R, some Ip S is divisible by R. Hence 8 is divisible by R, and as the coefficients in 8 are relatively R. We suppose the sign of R to prime, we have 8
respect to initial
of
R
be fixed according to any suitable convention, and treat as unique.
We shall call R the resultant We shall now prove that U is that U is linear in y. Let
c?i
be the
of
A
and B.
of order
in
y and, indeed,
obtained by adjoining x to 3\ (1) in Si, using a w defined by
field
form a resolvent for
=
w
(6)
ao
We
+ py,
=
with ^ a rational function of x. Let the resolvent be V and JV forms in the a*, 6$, and and let y N/M, with
=
M
te;.
The system A, B,
(7)
equivalent to the
is
aQ
iv
py
system
(8)
A
solution of the
My ^=0. result
by
0i
from V, OI
first
If,
M,
two forms
then,
F
A
,
Jtfi,
N respectively V^M.y
will be a basic set in
^
TI
-ZVi
are
F= 0,
the forms which
on replacing
w
by a
and
+ P,
(9)
in
in (8) will satisfy
which hold
for
5
N, X
,
the totality of
all
forms
(1).
cannot be of higher rank in a than R. This that and NI are of lower rank in OQ than JB. J/i implies Let M^y NI be written in the form Evidently
with the
&
coefficients
"Pi
and T, forms in the ca and and with a a polynomial in x.
6i,
with integral
ALGEBRAIC DIFFERENTIAL EQUATIONS
50
The numerator
(10) holds 2.
By Sj^O.
in
18,
Then Let j be such that a non-zero form of 2 reduced with respect to J2.
holds 2.
that U, in
A
Thus, for
SiyTi
SjyTj
is
This proves
linear in y.
is
(5),
each
B
and
have a common solution in y,
to
it
is
necessary that OQ,
dn\ VQ,
,
'
'
'
9
Om
of the resultant of A and B. and a solution, if it does not annul If a certain fixed form in a &m,* then A and B have a single solution in common, which can be expressed rationally b m with integral coefficients.^ in terms of a be a solution in the general solution -
b m is such
-
(to,
,
,
,
,
,
,
prove now that the resultant of A and B is a linear combination of A, and a certain number of their derivatives, the coefficients in the linear combination being forms with
We
B
integral coefficients.
In
A
and
a
let
R,
-a
aiy
be replaced respectively by
bQ
n
B
Pn,
6t
bm
y
Qm,
B
R
and their be expanded as a polynomial in A, The term not involving A, B, or their derivatives, As we will be a form in the unknowns (3) which holds (1). and
let
derivatives.
saw, such a form vanishes identically.
The methods
of
V
Chapter
This gives our result.!
permit the actual construction
of resultants.
ANALOGUE OF AN ALGEBRAIC THEOREM OF KRONECKER 35- It of
is
a theorem of Kronecker that,
equivalent
given any system
n unknowns, there exists an We system containing n + 1 or fewer equations. equations
algebraic
in
present an analogous theorem for differential equations. *
The
coefficient of
y
in
U.
We
are using the expression "single solution" in the sense of analytic function theory rather than in the sense of 6.
t
JFor a theory Journal
ftir
die
r.
of resultants u, a.
of
linear
Mathematik,
KOnig, Algebraische Grqften,
vol.
p. 234.
differential
116 (1896),
forms, p. 157.
see Heffter,
APPLICATIONS
III.
51
contain a non-constant function. Let 2 be any system of forms in y l9 Then there exists yn a sijstem Q>, composed of n-\- I or fewer forms, ivhose manifold
THEOREM.
$
Let
.
,
with that of 2.
is identical
If 2
consists
of a
number
finite
of non-zero forms *\, ..-,*;,
(ii)
then a system
Q)
which
exists
is
composed of I\ and n or fewer
combinations, ivith coefficients in shall need the following lemma,
linear
We
F
of which
3r,
2
F
-
,
r
,
applies
.
to
a perfectly general field. LEMMA. Let be a closed irreducible system, mui,--',u q u q a se t f arbitrary unknowns. yi> ypj with u
W
'
\
'
'
>
Then
in
>
j
there exists a basic set for
ivhich, if
an AI
involves a HJ effectively,
of Ai, with respect
vative
W
the partial
to the highest derivative
deri-
of Uj in At,
does not belong to *P.
We
show
^
of
of
Prom rank
how
first
class #
+ 1,
to
we
choose AI From among all forms select those of least rank in y la
the forms just selected,
.
we choose such
as have a least
u qj and continue, taking the ranks in M^-I, MI, in succession, as low as possible. For A L we take any of the forms thus obtained. Obviously AI fulfills our requirein
,
,
V
of class In choosing A 2 we first take all forms of 2 which are reduced with respect to AI. From these, select such as have a least rank in y* and continue as
ments.
#
+
we
above with respect to the m. In the same way,
We
find thus
we
determine A*, to meet the requirements of the lemma.
,
an
Ap
,
A
2
as specified.
in
succession,
36. Returning to the proof of our theorem, we limit ourselves, as, according to 7, we may, to the consideration of
the ^2
finite
9
>
vr
1 new unknowns, (11). Introducing r consider the system 21 composed of the two
system ,
we
,
forms (12)
F
ALGEBRAIC DIFFERENTIAL EQUATIONS
52
Let
be used to represent the system (11) when the un-
,&!
knowns are the Let >/!,
We then
-5*1
-,
y*, v*.
be resolved into closed essential irreducible systems 9 Suppose that 1 does not hold some A, say .///.
A
.
1 unknowns among yly yn , say that, given any n 1 in those unknowns contains non-zero forms n .//; ,
and the
For
Vi.
instance, suppose that./// does not contain
a non-zero form in '
(13)
Vt,'-,V r
*/n-r,
',
2/1,
.
(13) will be a set of arbitrary unknowns for .///, so that 4j will have a basic set consisting of one form, B, take which introduces yn algebraically irreducible. is the manifold of .///. Then the general solution of
Then
B
We
.
B
We
prove that
shall
B
instance let
involve
does not involve the
all
arrangements
of
we have treated the unknowns us now give them the order
and
solution,
B
as
any
is
For
if
followed (13).
yn
we vary
If
number some point
finite
the same
Thus
the unknowns.
Consider any regular solution of B. this
Vi.
ty.
22, the general solution of
According to manifold for
B
the yi in
derivatives
their
of
far
Let
we
can, using the vr so determine given regular solution, B.* But this cona of to solution as second regular get is free of the v^ Thus tradicts the fact that J\ holds Jj.
arbitrarily,
vt
,
,
but slightly, at
?,
v r -i of the
B
given any solution yl9 yn in the as a of form solution considered in the general yi alone, and given any analytic functions t;2 , vr the given yt f Vi constitute a solution of ^//, hence a solution of
This
mfeans
that,
,
B
,
t;.
* Of
course
we have
to
*!
+
,
+ tv*V.
construct
new
analytic functions yi which
assume, with their derivatives, the modified values at
{.
III.
APPLICATIONS
53
F
F
But, as the vi can be given arbitrarily, r must 2 , vanish separately for the given y*. This means that ,
H
holds AJ %
Let
Our statement
Al9
Aq
,
Consider any n
We
proved.
A
be those 1
of the
extract from each
in the yi of (14)
is
y,-,
A^
and the
which are not held by &i.
t
i
=
Vi.
1, #, a non-zero form Let the 3 forms thus obtained ,
be multiplied together. We obtain thus, for every set (14), a form which vanishes for every solution of 2 l which is not a solution of
we can
Q
Now, as of contains non-constant functions, so fix the Vi in of that every form obtained above .
l
becomes a non-zero form in its set (14). Let the system of forms thus obtained, from the various sets (14), be denoted
by
Let /7i represent the system of two forms in the t/ alone which 2} becomes when the vi are fixed definitely as above. Then every solution of #t which is not a solution of 2 is a solution of (Dlt
The unknowns
Vi,
whose
We
from our discussion.
new unknowns by adjoining
Let
2
the
yt,
episodic,
n
L
.
We
now disappear 1 introduce r
wr
and consider the system 22 obtained Hi, the form
w*,
to
was
role
examine
,
be used to represent 2, considered as a system in Let 22 be decomposed into closed essential wi.
irreducible systems
A^
J
8 2 does not Suppose that hold some A, say Aj. We say that, given any n 2 of the yi then Aj contains a non-zero form in those n 2 y< and the wi. Imagine, for instance, that Aj does not contain ,
.
9
a non-zero form in yl9 -, yn -z and the wt. form of O^ is in Aj. For, let Q be any form of
they annul
Fk
.
If the yi
do not annul
Fk
,
then, since
ALGEBRAIC DIFFERENTIAL EQUATIONS
54
they annul each form of IIl9 they must, as seen annul G. Thus GFk holds A^ so that G is in Aj.
above,
Thus Aj has a form in any n 1 of the unknowns yi. Hence y and the a set of arbitrary unare n -z wt y knowns for yfjj and Aj has a basic set Bl9 J92 which int
,
,
,
B
B
troduces y n -i and y n respectively. Let 1 and 2 be taken, as in the lemma of 35, so that, if one of them involves
a Wi,
derivative with respect to
its
the highest derivative In addition, let J? t be algebraically
of that wi is not in Aj. irreducible.
We
For instance, that B^ involves Wk effectively. Let G be the form suppose of
,
in the general solution of
the
unknowns
the
-
BI
in
y n-i
in
Blu *
This, cannot be, for, ordering
wu comes
that
so
we
last,
regular solution of
B
find that
j/i any can, together with any finite number of their derivatives, be given slight, but otherwise arbitrary, variations, at some point and ,
,
,
,
,
WH then be determined for a second regular solution of BI. Let S be the Again, suppose that B2 involves Wk*
B
2 with respect to the highest derivative of Consider a B* 2 for which regular solution of BI 8 does not vanish. Let if be the form of i in yl9 y n -2 be a point for which the functions in the yn alone. Let
derivative of
Wk
in
.t
,
B
,
solution and the coefficients of
H
BI
B
2
,
,
H are
9
analytic, with
and for which neither 8 nor the separants and initials of BI, B2 vanish. We can but arbitrarily at and deter-, yn -
not
all zero,
,
-
-
-
,
any at
finite ,
of its derivatives slightly,
determine Wk from
solution of
number
of
BI,
jB8
.
B
2
= 0,
Thus we
but arbitrarily,
securing another regular can get a regular solution
BI, Bt which does not annul H. *The remainder t Notice that
S
of is
G
with respect to BI holds Aj and thus
not the separant of
in their original order.
B
t
.
We
are using the
is
0.
unknowns
APPLICATIONS
III.
Thus
w2
functions
B
and
J5i
a regular
in
-,
,
F
wr
B i7 B
of
2
with
,
a solution of
give
,
Then the yl9
WL
are free of the
2
solution
55
arbitrary
22
.
,
yn
analytic
This means
hold Aj so that since FI, as a form of holds ^//, This contradiction proves holds ^//. 2 2 of the y,-, and the Wi. that Aj has a form in any n v/ 7 be those systems A{ which are not held Let
that .F2
52
,
9
,
^
by
all
r
,
,
Consider any
.
2
-
.
,
n
2 of the
i=l
We
y<,
extract from each A^ q a non-zero form of the and in (15), multiply together the q forms thus yt obtained. get, for every set (15), a form which is 9
,
We
annulled by every solution of 22 which is not a solution of We fix the w in 3? so that each of the foregoing 2 forms becomes a non-zero form in its unknowns (15). .
Let
2
be the set of forms thus obtained.
the system which
Then every
solution
a solution of
We
22 becomes when
cf>
2
of ZT2
which
Let //2 be
the Wi are fixed as above.
is
not a solution of
2
is
.
form a system
23
,
adjoining to //2 the form
where the Zi are unknowns. We introduce s in the expected way. Let Aj be a closed essential irreducible system held by 2> 8 which ^ 8 does not hold. We have to show that, 3 of the y^ there is a non-zero form in Aj given any n in those yi and the zi alone. Suppose that Aj does not contain a non-zero form in yi, yn-s and the #. Then, as every form of
set
be
selected
as
in
lemma
the
of
35.
Furthermore,
abbreviate the proof, let us assume that BI is merely see at once that B^ involves algebraically irreducible. to
We
no
if
jB 2 involved
Also, slight variations on the yi, Zi.
a ,
we could practice arbitrary _ yn 8 yn-\ and their derivatives z/,
,
5
ALGEBRAIC DIFFERENTIAL EQUATIONS
56
Si J?2 and get a second such regular This cannot be, since every such regular solution would have to be a solution of the form of <2>a in yi, yw -8,
in a regular solution of
,
solution.
,
S
if
we
involved a
could take yi,
Zi> Finally, yw-8, of and finite number derivatives, quite arbitrarily, y any at some point, and get a regular solution of Si J5a J58 This contradicts the fact that
y/i-i.
B
,
rt ,
,
.
,
alone. z\ being properly fixed, we get two systems, the latter containing a non-zero form in every 3 of the y,-,.such that II3 holds 2 and that every solution
Thus, the J78 and
n of
7/3
,
which
is
not a solution of
2
is
a solution of
3
.
In /Tg, there are four forms.
Continuing,
we
a system equivalent to 2,
find
containing
at most n-\-l forms.
That w nection
+1
equations
may
with n unknowns,
is
actually be necessary, in conseen on considering the system
in y,
y?
4y,
2>
2/ 2
which defines the general solution of yj 4y. If a single form, G, had the manifold of this system, G would have to be of the first order in y. Then G would have to be divi-
=
sible by y\ 0, 4y, and so would admit the solution y which does not satisfy the given system. 37 The assumption above that $ does not consist entirely of constants is essential. For instance the system in y,
vu has no solutions.
Still for
-y*>
y
i
any pair of constants
di
and d8
,
the form
diy'
+d
2
(y
1)
has solutions in common with y^.
However, the following result, which can doubtless be improved, holds for fields of constants. THEOREM. Let A be an algebraically irreducible form in a single unknown y,
the
order of
A
in y being r.
Then
APPLICATIONS
III.
57
there exists a system of forms, consisting of A and of at most r other forms, whose manifold is the general solution of A. We shall need the following lemma:
LEMMA: Let 2i9
28
none of which holds any other, and let 2 be a closed system which holds no 2i. Then there exists in 2 a form which holds no 2i. ,
be closed irreducible systems,
We proceed by induction. The lemma is true for s = 1 We shall prove that the truth for s 1 implies the truth
.
for s.
has
a
i
1,
=
The
truth for s
implies that each
1
2i9
i
=
1
.,
,
s,
form AI which holds no 2j with j \ i. Let Bi, 2 which does not hold 2*. Let -, s, be a form of JTi
AI
A.\
i
AS,
A.i-^\
L
'I
y
,
S.
Consider the form C^
which belongs Pi for no 2i.
Let
i
>
A
1
to 2.
holds
/?
~P
'-.:
-5"i ,
7?
Since PijBi does not hold 2l9 and since C does not hold 2^ Similarly, C holds .
be resolved into closed essential irreducible systems,
2 219 9
2
~P
-I
I
2a
9
,
having the general solution of A for manifold. Let Bi, s be a non-zero form of lowest rank in 2i so 1, that the manifold of 2i is the general solution of Bi. (We may and shall assume that each Bi is algebraically irreducible.) Each 2i is held by the separant S of A, hence by the resultant i
=
,
9
(as in algebra) with respect to
polynomials in
y
r
.
than r in y. Let AI be any form of
examine the system
yr
of
A
and
This means that each
A A 9
1
2 .
S
which holds no This system
considered as
J?< is
is
of order less
We
2t.
shall
equivalent to the
set of systems
Let
A
l
d be the remainder of AI
does
not
hold
2
t
,
d$Q.
with respect to Bi.
As
d
holds 2i
Because
+ Al9
the
resultant of Bi and Ci with respect to the highest derivative
ALGEBRAIC DIFFERENTIAL EQUATIONS
58
Because Bi is algebraically irreducible Bi holds 2{ -f A l and of higher rank than Ci, this resultant is not zero.
in
.
This means that
if
A, A^
irreducible systems,
resolved into closed essential
is
2 2 *9
'
'
'>
9
each
will contain a non-zero
2'i
Choosing now
a form
A2
in
2
't>
form
2
1.* of order less than r which holds no 2'it we form
A l9 A 2
and operate as above. Continuing, we after adjoining, to ,^4., r or fewer forms of 2, we get a system of forms whose manifold is that of 2. the system A,
find that,
FORM
QUOTIENTS
A
An
38. 2/i,
B
are forms in and expression A/B, where not identically zero, will be called a/orm yn with
-
,
B
,
Two
form quotients will be considered equal if It is easy are as rational functions of the y
quotient.
and
sufficient that
given analytic yl9
they yield the same analytic function for -, yn which do not annul BD.
Let
y
(16)
where
A
and
B
=
are forms in
question which we shall study is y in the case in which u
to
^, The a single unknown u. a that of attributing meaning is
such that both
A
and
B
vanish.
2
The totality solutions of
of forms in
for all
By-A =
(17)
B$
y und u which vanish
an irreducible system. The manifold of 2 is the general solution of the equation obtained on dividing (17) considered as by the highest commoji factor of A and with
is
B
polynomials in the int. *If r
f The
= 1,
this
results
of
means that there are no S. Chapter VI will show that the manifold involved
independent of the field employed.
is
III.
A
APPLICATIONS
59
function y will be said to correspond if M, y is a solution of 2.
to
a function u
through (16)
Example
Then every
Let
1.
analytic
y corresponds
to
u when u
= 0.
For,
be any analytic function, not identically zero. If k is ka. Allowing k a non-zero constant, y a^la when u
let
=
=
approach zero, we find manifold of -2.* By taking
that
to
0,
suitably,
become any desired analytic function Example 2. Let
=
v y
i/
we
belongs to the can make i/
some
(in
area).
.
u
12, we see that, since UL does not Referring to Example 2, vanish for every solution in the general solution of uy u\, 2 f or u in the must. then */ 2w Thus, general y 2 ?*!*/! 0. If u k, y approaches zero solution, we must have y
+
=
=
Thus y
uniformly as k approaches zero. function, corresponds to u
Example
We
Example
We
find,
4.
no y corresponds to u Let
y
=
putting y
1
U Ui
(1 8)
Differentiating, (19)
(20) *
P
=
(ttJ
=
= 0.
+ U,) (!+) that
z,
Z
U\
(!+ W) + U\
we have u\e
where
= 0,
Let
3.
find that
= 0.
=
+ uu
l
el
=
2 w8 (ui+
See next to last paragraph of
22.
u2 P,
.
and no other
ALGEBRAIC DIFFERENTIAL EQUATIONS
60
we
Multiplying (19) by uz, and using (18), (21)
ulg
where
We
=
Q
(22)
u
z
+ uulg = 1
P
u2 (u,
Q
+ u gj (M! + u).
multiply (19) by HI and subtract from (21).
MiP)
i(Q
Then Q
UiP
ul
P=Q
=
Then
0.
holds 2.
u
Suppose that
Q
z
u2
find
a constant
is
Q==0.
implies
=
=
it
By
distinct (22),
from 0.
zP=0,
=
Then
so that,
2 If we let and y 1. Thus * 0. by (20), uz I M k-\-hx, y approaches 1 as h approaches 0. Thus y and no other function, corresponds to u when u is a non-
=
zero constant.
= A,
the other hand, if we let u as h approaches 0. proaches 0. function y corresponds to u
On
(!+)/
=
=
we find that y apThus every analytic
As the general solution of an equation is completely by a finite number of algebraic differential equations we see that, if u makes A and B vanish, then either every 39.
defined
y which correspond to u are the totality of solutions of a system of algebraic differential equations, in the coefficients of which, u figures. We are going to study the circumstances under which no analytic
y corresponds
y corresponds Writing
A
to a
=
C(v)
(23)
to
else,
the functions
u which annuls
A
and B.
=
J3(u),
we
let
A(u + v),
D(v)
=
B&+V).
A(u),
=
u, or
B
C and D are forms in which are not necessarily in vanish. Let
Then
v, r .
with analytic coefficients makes C and Also, v
=
D
C= E
F
E+H,
D = F+K, D
where and contain respectively those terms of C and which are of lowest total degree in v and its derivatives.
III.
APPLICATIONS
61
E
is at least that of F. Suppose first that the degree of when Let be an analytic function which does not annul substituted for v. Let v be replaced in C and D by ha, where h is a constant. As h approaches 0, C/D will approach uniformly to an analytic function, which corresponds to u through (16).
F
E
=
corlower degree than F, we see that y B/A. In that case, we shall responds to u through y oo corresponds to u through (16). With this say that y vanish has at least convention, every u for which A and one corresponding y. There appear to be grounds for conjecturing that if u anIf
is
of
=
=
B
nuls both w",
A
and B, and
if
more than one y corresponds
then every analytic y corresponds to u.
to
CHAPTER
IV
SYSTEMS OF ALGEBRAIC EQUATIONS 40. The preceding chapters contain, of course, a theory of systems of algebraic equations in n unknowns, with anaOne has only to suppose that the given lytic coefficients.
2 consists of forms which are of order zero in each But there are good reasons why algebraic systems should
system yi.
receive special treatment.
To begin
with, in most of the foregoing theory, algebraic are forced into an artificial association with diffeequations For instance, the closed essential irrerential equations.
ducible systems held by a system of forms of order zero in each ?/i, are systems of differential forms. One does not
obtain thus Kronecker's theory of algebraic manifolds.
We
shall see that a purely algebraic theory of algebraic systems can be secured with the help of the notion of relative irre16. which was studied in But what is more important for us, from the standpoint
ducibility
of differential equation theory, is that the theory of algebraic systems can be developed from the algorithmic point of view, so that every entity whose existence is established is
constructed with a finite number of operations. The reof the algebraic theory, when applied to systems of
sults
differential forms, will give us
methods for determining the
basic sets of the irreducible systems in a decomposition of a given finite system of differential forms. A theoretical
process will be given for obtaining equations which comAlso, we shall be pletely define the irreducible systems. able actually to construct the resolvents of irreducible systems of differential forms.
62
IV.
Finally,
we
ALGEBRAIC SYSTEMS
63
shall apply the theory of algebraic systems to
the study of the organic properties of the manifolds of systems of differential forms.
Our results relative to algebraic systems are mainly contained in the literature on algebraic manifolds.* For us, it will be convenient, in deriving these results, to use the methods
of Chapters I
and
II.
INDECOMPOSABLE SYSTEMS OF SIMPLE FORMS 41.
We
define a
domain of rationality
to be, as in algebra,
a set of elements upon which the rational operations are performable, the set being closed with respect to such operations.t Every field is a domain of rationality.
Let a domain of rationality 3) be given whose elements are functions of x, meromorphic in a given open region 21. simple form, we of order zero in each
By a is
mean a form in y^ y n which Wherever the contrary is not iji.% ,
,
the coefficients in a simple form will be understood
stated,
to belong to 3). system 2 of simple forms will be said to be simply closed
A
every simple form which holds 2 belongs to 2. A system 2 of simple forms will be called decomposable such that neither there exist two simple forms G and if
H
nor
H holds 2,
while
QH holds 2.
A
system which
is
if
O
not
decomposable will be called indecomposable.
2l
Every system 2 of simple forms is equivalent to a finite set 2S of indecomposable systems. This is proved as in and, in fact, is an immediate application of the results 13, ,
,
*
Macaulay, Modular Systems.
Van
der Waerden,
Moderne Algebra,
vol. 2.
t The term ''domain of rationality" is being displaced, in common usage, by the term "field". We have reserved the latter term for use as in the preceding chapters. See Dickson, Algebras and their arithmetics, Chapter XI. J We prefer this term to polynomial, since we shall have to use the
term in more general situations. One can use here Hilbert's theorem on the existence of a finite basis every system of polynomials in n variables, in place of the lemma
latter
for of
7.
ALGEBRAIC DIFFERENTIAL EQUATIONS
64
on relative irreducibility unique in the sense of
The decomposition
16.*
in
is
14.
SIMPLE RESOLVENTS
2
42. Let
-iVpj free of the 2/i
>
2
form of
P+Q y^
in yj
while,
for j =
and the
Ui
,
two
into
u\
sets,
,
uq and
,
suc h that no non-zero form of
ni
1,
-,
itq
,
p, there
We
ui alone.
is
y\
;
,
shall call the
,
yp
2
is
a non-zero
Let the unknowns be
of unconditioned unknowns. the order
and
Then
be any non-trivial simply closed system.
the unknowns can be divided
m
a set
listed
in
,
let
A 19
(1)
be a basic set of 2.
Then
evert/ solution
vanishes
is
Each of
a solution of 2
.
(1)
...,A P
^
introduces yt.
/or w/uc&
Furthermore,
fe
of no Ai
initial
if 2 is indecomposable,
then (1) has regular solutions and every simple vanishes for the regular solutions of (I) is in facts are evident.
form which
23
These
be a non-trivial simply closed system. We are going to show the existence of a simple form (?, free of the yi and of a form 43.
Let
2
Q
=
lfi*/iH
where the Mi are simple forms free of the y^ such that, for two distinct solutions of 2 with the same m (if m exist), and with G^O, Q gives two distinct functions of x.
By a prime
system,
we
shall
understand a simply closed
indecomposable system. 25, we consider the system of forms obtained Following from 2 by replacing each yi by a new unknown Zi. We take * first
To
satisfy all formalities, one can take, as c?,
the field obtained by
adjoining to 3) the derivatives of all orders of the function sin
and then forming larged
all
rational -combinations of the
functions
in
set.
t The definition of regular solution
is,
of course, that of
23.
3),
the en-
ALGEBRAIC SYSTEMS
IV.
&
the system Zi
composed of the forms and also the form
65
of 2, the
forms in the
just described,
which the
A
are unknowns.
Let be any prime system not contain which *<. holds, and which does every form yi shall prove that contains a non-zero form which in-
in
A;
&
A
We
unknowns other than the
volves no
m
and
fa.
A
contains a non-zero form in the ut alone, we have our result. Suppose that contains no such form. If
A
A
Since
has
all
A
forms in 2,
for j
=
1, p, a non-zero form Bj in yj and the alone. Then Ij the initial of Bj, since it involves only the m, is not in A. in 1, Similarly, let Cj, j p be a non-zero form of
has,
,
m
=
m
and the
Zj
alone.
A
,
follow the Ui in C/,
Letting Zj is not in A.
we
see
that the initial // of C;
To
fix
our ideas,
let
us assume that y^
Consider any solution of
z
is
not in A.
which
for
(yi-*i)/i-"lpli'---/,!
(2)
does not vanish.
_
2
AI
Let and
A
m
For such a ^g (y
solution,
^2)
we have
H-----h lp(yP
*P )
Viti
be the maximum of the degrees of the Bj in the yj be any positive Let
of the degrees of the Cj in the gj.
We
integer.
where
F
5 is
write, for s
=
a simple form.
=
=
-,,
0,
Now,
plain that, using the can depress the degree of it is
0, we 0, Cj Bj F* in each yj and in each gj to be less than m. expression for each Af will be of the form
relations
3* 1
_ ~
Ea
The new
ALGEBRAIC DIFFERENTIAL EQUATIONS
66
D
where
8
a product of powers of the 7/ and Ij. Let write s multiple of the
is
s
= 0,
,
m
D
common
be the least
H
each
,
9
D
We
.
being a simple form of degree less
Now
the number of power products than m in each y. and #. is w 2p degree less 2 Consequently, if we take a 2> w ^, we can find a non-zero polynomial in ^, of degree not greater than a, whose co-
than
each
in
and
y.
z..
of the y., #. of
.
efficients are simple forms in A 8 which *, Aj, and the M*, vanishes for every solution of for which (2) does not vanish. thus obtained belongs to A. The form in the A*, ,
A
m
The existence
of
and
Q
is
then proved as in
25.
We
we are dealing with simple forms, it is possible to take the Mi here, which correspond to the ^ of 25, and to the Mi of 26, as integers in short, no derivatives
notice
since
that,
;
of the %i will
2
25. appear in the forms K, L of be any non-trivial prime system.
44.
Let
We We
take a pair introduce a
(?,
Q
as in
43.
a system A by ^ be Let the Q system of all simple adjoining forms in w, the u i and y. which vanish for all solutions of A.
w
new unknown w, and form
to 2.
28, that easy to prove, as in 12 Those forms of which are free of It
is
is
w
indecomposable.
are precisely the
forms of 2.
As above, we prove that the y t
We
now arrange
unknowns
the
u l9 "- u 9
and take a basic set for
A
(4)
Here, w, y l9
We
has a non-zero form free of
.
take
,
A
,
q;
w\
Q
in
yv
in the order
--,yp
& AI y
,
Ap
yp are introduced
.
in succession.
algebraically irreducible relative to 3).
IV.
As
in
follows that each
it
29,
ALGEBRAIC SYSTEMS
=
A
t
67
yi so that
linear in
is
the equation A i expresses y i rationally in w and the Uj. a simple resolvent of 2 (or call the equation A
=
We
of
of simple
any system It
is
now
easy
manner
of the
in
forms equivalent to
to prove that q,
which the
m
in
-2).*
42,
is
independent
are selected.
BASIC SETS OF PRIME SYSTEMS 45.
We
unknowns
consider simple forms in the
Let AI, A*,
(5)
-
-
-,
Ap
be an ascending set of simple forms, each Ai being of class are going to find a condition for (5) to be a basic
We
q-\-i.
set for a
prime system. In what follows immediately, we consider the to be x. and and the to be functions the of u complex variables, yi i
m
We
represent Ai, with this interpretation of the symbols in
by
it,
at.
We
denote
MI,
,
uq
93
by
an open region which x
in
for every point of
,
the space of x\
lies in 21.
We
are going to prove that, for (5) to be a basic set of a prime system, it is necessary and sufficient that
Given any open region
(a)
of x; u ly in
-
,
uq
,
93,
there exist p functions,
(p)
",
',
,
analytic in some open region contained 04 vanish when they are substituted
which make each
93,
for yi,
-
,
i
(b)/0r every
of x\ Ui,
yp
and
',,
~
1} (i C given any analytic functions a t -i to vanish when u q which cause i,
substituted
respectively,
,
for yl9
,
yi
i,
the
coefficient
of
the highest
= t^, j
'
power of yi in ai does not vanish for i
,
in *
yi,
One
is
1 is
yj
1,
and, after these substitutions, Oi, as a polynomial irreducible in the domain of rationality obtained ,
equipped
now
to read the first part of Chapter VII.
ALGEBRAIC DIFFERENTIAL EQUATIONS
68
by adjoining
tions?, --,
to 3) the variables MI '- 1) (t
,
,
u q and
the func-
.*
Furthermore, we shall see that if (a) and (b) are fulfilled, no non-zero polynomial in the m, yt, with coefficients in 3), of lower degree than each a/ in yj, j 1, p can vanish 46. In
We
1,
,
^)
we
46, 47,
,
where the t^are functions
as in (a).
treat the necessity of the conditions.
assume the existence of a prime system
a basic
is
=
f^, j =
for yj
2
which
for
(5)
set.
Let mi be the degree of at in yi. Let 3> be the domain of rationality obtained by adjoining the uj to 3). If 0,1 were reducible in 3) would be the product of two forms, U and V, each of degree less than mi in y^. As one of (7, F would hold -2, (5) could not be a basic set. ,
Then
a\
is
^
irreducible, so that the equation ai
mines y l9 in some open region 33i, contained one of mi distinct analytic functions i,
w? variables #; MI, As the coefficient of the
=
deter-
in 23,
as
any
m
of
the
,
t
.
,
lower degree
in
y
highest power of y% in a2
is
of
than a 1? that coefficient cannot vanish
l
u q if y v is replaced by any J. be replaced by some & in a 2 and let # 2 be the Let 3)* be polynomial in y* which is thus obtained from a2
identically in #; MI,
,
^
Let
.
the domain of rationality obtained on adjoining the indicated Cl to
3V
Suppose that 2 as a polynomial in y% is reducible in 3)t of positive Let 2 ?>! y 2 with (pi and y a polynomials in y 2 degree, with coefficients in 3)*. Each coefficient in y>i and $p 2 where d and /^ are polynomials in is of the form d/f, .
,
=
u i9 Let
-
9
Uq\
,
Ci
with coefficients in 3).
be the product of
HI in
,
denominators
=
6a,
(6) *
all
We
ft.
may
write
ViVt
Thus, the enlarged domain of rationality is a set of functions of a?; as a polynomial For * 1 the above is to mean that a\ ,
1/1,
is
<,
.
=
irreducible
,
,
when
Ui,
,
w7
are adjoined to 3).
The above result establishes an equivalence between the basic sets of prime systems and certain sets of polynomials used by van der Waerden in
ALGEBRAIC SYSTEMS
IV.
where whose
and
w
are polynomials in y2 of degree less than 2, coefficients are polynomials in HI, u q C{. Making
^
t
i/> 2
use of the relation
&
in
69
of
the
,
=
i
in
coefficients
for
CJ
I//!,
V> 2
we
\
depress the degrees
Multiplying through in (6) by U of the denominators y, one obtains a relation
Ma =
?ii
a
with
and ?2 polynomials in
?i
coefficients
m
2
.
We
in
t
w
degree less than
of
t/ 2
are polynomials in {; MI, -, u?, notice that neither ?i nor J 2 {.
of % l9 ? 2
of degree less than
Each
.
the product
ui,-,Uq.
The
l
form y/y, with y a polynomial in
coefficient will be of the
(7)
m
to less than
u q y*. Let t,g\jg* be vanishes identically in x\ ui, the polynomials which result respectively from ^0, ?i, J 2 <
\
t
on replacing
J
by
Then
y\.
ta 2
(8)
=
Let vanishes for y l t|.. We obtain a relation
g l g2
s^(ta2
(9)
be the coefficient of
s1
)ka = l
with k and 6 polynomials in degree less than mi in y^.
Now
b
MI,
-,
y^
b
uqj
and
y\, yz,
of
b
=
,
=
As
fi.
ai is
irreducible in
the coefficients in b are of degree less than b must be identically zero.
>
m
as
Ji
.
vanishes identically in ui, CJ. u^; 2/2 for yi b is written as a polynomial in z/ 2 each coefficient
Hence, if must vanish for y L
Let
ax
in
be the
forms which
t,
as unknowns.
g Then 2
,
R(TAt
(10)
We observe A 19 A 2 and
g
A 1 and let T, GI k become when the m,
initial of l9
O
that GI and are not zero.
his treatment of
,
l
G*)
G2
KAi
G
2
,
K
yt are
and
ylt
be the
regarded
.
are reduced with respect to (10) shows that l G 1 G 2
Now
prime ideals of polynomials.
vol. 96, 1927, p. 189.)
=
,
x
in
(Mathematische Annalen,
ALGEBRAIC DIFFERENTIAL EQUATIONS
70
Thus one
holds 2.
of
G l9 G
must hold 2, which
2
im-
is
possible.
Hence 2 is irreducible 47. Thus the equation functions
^ =
/',
1,
$
2
=
2
w
-,
some open region
in
2V
in
2
defines
of
,
MI,
a?;
y t as one of w2 each analytic -, u qj
Evidently we which will serve no matter
contained in $1.
,
can use a single open region 93 2 which & is used in determining the m 2 functions f". That is, we have m, m 2 pairs &, j' which are solutions for ^i, 1/2 and these m^ w 2 pairs are the only solutions of i 0, a t
= = = ^ 0, a =
of
analytic in 2S 2 .*
2
Consider any pair &,
=
Let
is
2
than
w
2
is
ss
does not
we
^8
= K-
becomes when y
see, since s 3
is of
degree less
,
=
But those coefficients are of lower degree This proves our statement. be obtained from by the adjunction of } and C; ^/ be the polynomial in //3 which as becomes for z/ x K, fj.
.
3
=
ty.
We
shall
irreducible in
Suppose that nomials in
Each
which
i/2
show that
Suppose that s3 does vanish.
t".
r
Let ==:
be the coefficient of the
s3
shall
that the coefficients of the powers of */ 2 in tf That is, the coefficients of the powers of yt in ss
in y%
are zero.
Let
=
irreducible in 3) x-,
vanish for y than mi in yi.
2/a
We
.
be the polynomial in
<5
As
Let
fj'.
highest power of ys in a3 vanish when y\ &, 2/2
2/3
prove that
#.
Then
it is not.
of
3
degree
coefficient in
less
y t and y 2
3
than is
=^^ m
s
of the
a polynomial in y3
as
,
with
2
^ and
poly-
y> 2
with coefficients in
,
form
d/ft,
where
rf
#.
and ^
are polynomials in z^, ,%; f{, fj', with coefficients in Let be the product of the denominators ft. Then
Oa B
(11)
=^
Making use
the relation
of
*
The
sets of
m
ip l
,
functions
have functions in common.
i^ 2 C}'
to
2
3).
i// 2
where Vi ^nd t//2 are polynomials in y$ of degree whose coefficients are polynomials in MI, each coefficient in
,
=
for ff
less ,
,
Mg
we
than C{,
;
iw s
,
Cj'.
depress
be of degree less than
corresponding to two distinct
C<
m
2
may
IV.
in
The new
".
ALGEBRAIC SYSTEMS
be of the form Yin with
will
coefficients
71
a power of the coefficient of yf* in 2 and y a polynomial *l m2 in ". Thus in M A -, u q C{, j' of lower degree than ,
if
0j
;
the highest of the powers
is
17,
we have
2 polynomials whose degrees in y$ / are respectwe m w the same In lower than B 2 depress the way ively to be less and in the coefficients in of ? 2 1 J degrees than Wi. We find thus
with fi
,
,
.
,
with
yi
a power of the coefficient of
2
in
a l9 and with
, j'; y 3 whose degrees u^; r tn 2 m^. in 2/3, Cj , i are respectively less than Further3 more, neither of r l9 r 2 vanishes identically in
polynomials in w l7
r lr rg
-,
w
Let
^,
^2 result respectively
^,
K by y and
replacing
t
Cj'
=
by y2
,
from
,
2
t
0, TI
and T2 on
Then
.
=
Let ^ and 52 be respectively vanishes for y{ fj, y2 /' Then we have the coefficients of y^\ in a x and of yf* in a 2 a relation .
Q
(12)
with
b
kz,
Jcij
of
degrees
sfr (ta 3
g2
)h
y
}
and
t/2
=
ks a 2
cii
polynomials in M!, in
b
fft
uq
-,
being less
I
ylf y2 than mi \
,
y$j
the
and
w
2
respectively.
Now
u q y& if yi and y2 vanishes identically in w t are replaced by & and f" respectively. Hence, if 6 is written as a polynomial in yB each coefficient must vanish for == &f 2/2 y\ Considering the degrees of the coefficients fj'. m
b
,
=
in
i/!
,
,
,
:
and
t/2
,
we
see by the argument used in proving that
does not vanish, that 6 vanishes identically. a relation
Thus
$5
(12) gives
ALGEBRAIC DIFFERENTIAL EQUATIONS
72 (13)
/* (TA B - G,
/f'
0.)
-
JKi
^ K
=
A,
2
with 7t the initial of -4*, and with G\, G% not zero and reduced with respect to AI, A 2 A B This proves that 3 is irreducible in $/. -
.
,
Continuing, w e stated in 45.* r
We
48.
H
are simple forms such that of
vanishes for
Let nishes
which make no
(5)
all
',
,
when
conditions
Let the conto the sufficiency proof. 45 be satisfied. We shall prove that (5) which no initial vanishes and that, if G
has solutions for
lutions
the
of
necessity
now
turn
ditions stated in
and
the
prove
GH vanishes
initial
rr
(14) MI,
-,
?i,
;
uq
-
-
,
.
u) a-
are analytic, Ii va-
and no
= &, =
the (5)
become functions
with
...
/!
G
Let now
lp
and
+
remainders for vanishes for
all
Ip
G
x which
i
1,
,
constitute a solution of
0.
H
lutions with /!
of
va-
%q
which the
for
7;
Let
(i)
the coefficients in the Ai being analytic at nishing for the above values. If we take M* (i)
G
H
does. such solutions, or else (p} be functions as in Then no 45.
the yi are replaced by the
be values of x\
for all so-
then either
zero,
and
be such that
^
0.
Let
6?i
GH
vanishes for
and HI
H
all so-
be, respectively,
Then GI HI Then (?i JBi when the yt are
with respect to (5). Ip ^ 0. solutions with /i
must vanish identically
in
#;
2/ 1?
-,
uq
-
as above. This is because, if the quanreplaced by the tities (14) are varied slightly, but otherwise arbitrarily, the (i)
Ip ^ Hence give a solution of (5) with 1\ either GI or HI vanishes for the above replacements. Suppose that GI does. Then Gt being reduced wfth respect to (l>)
will
-
still
.
,
(5),
for *
vanishes identically.
which no
Thus
initial vanishes,
G
and
vanishes for
we have
One might replace the ahove proof hy an induction
>
all
solutions
our result. proof,
in
which
1 We think that, on proved for only one < with t the whole, the ahove treatment is less oppressive than one by induction.
irreducibility
is
.
ALGEBRAIC SYSTEMS
IV.
73
Let 2 be the totality of simple forms which vanish for the solutions of (5) for which no initial vanishes. Then 2 is simply closed, and indecomposable. Now, if 2 contained a non-zero form
reduced with respect to This p.
Cr,
=
u) i vanish for yi 1, Then (5) is a basic set for 2. ,
Of course,
2
,
contains no non-zero form
in
O
(5), is
would
impossible.
the
m
alone,
Also, by the methods of elimination frequently used, it can be shown that, for j 1 , p, 2 has a non-zero form ,
and the m alone. 49. Let (5) be a basic set
in yj
prime system 2. We have for which no initial vanishes,
of a
seen that every solution of (5) a solution of -5". We shall now prove that every solution of (5) for which no separant vanishes is a solution of 2. is
Consider any solution of
MU
(15)
which no
for
(5)
uq
,
y l9
\
Let
Si vanishes.
G
-,
yp
be any form in 2.
Let
XQ be
a point at which the functions in (15) and the coefficients in the Ai and O are analytic, and for which no
Let
Si vanishes for (15).
be the values
of
(15)
#
at
An
.
implicit function theorem shows that (p) as in ', 45, analytic at
easy application of the we can get functions
,
#o;
(16)
,*
1,
^
and assuming there the values If
we
put Ui
=
ut in C^,
make no
Since the f O) (17)
Mi
=
i?i,
?J,
we ,
i
=
i
zero,
xt;
as close as
values
/,-
.
=
.
,
,
1,
we can
i\
,
p respectively. ^,
t^ becomes
yj.
find values
Ji,---,li
please to (16), which, with the corresponding 0) make no 7* zero. If we take q'p of the ? ,
1,
which, with the Mi, no initial vanishes.
-,
0) give g, the S
^ analytic functions yj constitute a solution of (5) for which
Hence
G
vanishes for
ALGEBRAIC DIFFERENTIAL EQUATIONS
74
sufficiently close to x.
is
if a?o
By
continuity,
This means that G vanishes for (15).
Our
f
G
vanishes for
result is proved.
CONSTRUCTION OF RESOLVENTS 50. Before
we can
develop a method for the effective conwhich a basic
struction of a resolvent for a prime system for set is given,
we must have
a solution of the following problem.
Let A be a simple form in HI, u q w, of positive degree in w, irreducible as a polynomial in iv in 3> ( 46). Let AI be a simple form in MI, u q w\ y, of positive be function of x\ MI, uq in Let x degree any analytic y. ,
\
\
,
,
A
which renders
zero
when
substituted for w.
Let a be the
A. We
polynomial in y obtained by replacing w by 1 in assume that the initial of AI does not vanish for
w
=
determine the irreducible factors of
It is required to
the domain of rationality obtained
known
Several methods are
d
by adjoining
to S)
into its
for resolving
1-
in S>i, -
irre-
The following treatment is taken from van der Waerden's Moderne Algebra, vol. 1, p. 130, where ducible
factors.
a more general algebraic situation is considered. It nrist not be thought that we must actually possess to carry out the factorization. It will be seen that operations used
renders
it
Let
A
and that we get expressions a with no knowledge relative to fx except
are rational,
for the factors of
that
A
zero.
be of degree
analytic functions
render
A
zero
of
when
m
in
w
x\ Wi,
and
,%,
substituted for w.
be analytic in some open region 93. Let z be an indeterminate and let
i
=
2,
let
2
,
,
m be
other than
We
assume
l7
the
which
all
&
to
&
be the polynomial z fi which results on replacing y in a by y on m from result d replacing by &> 2, Let y &* ..., m
y and Let A, i in
& all
z
.
=
-
,
.
= AA
A
ALGEBRAIC SYSTEMS
IV.
75
the a polynomial in y, z with coefficients in coefficients being capable of determination by the theory of
Then
^
is
,
symmetric functions. Let y be resolved into irreducible factors This is possible, provided that we are able to factor in o.* Let a polynomial in one variable with coefficients in
=
Y
(18)
with each
=
a polynomial of positive
di
and irreducible
in
coefficients
di-.-dr degree in y,
with
z,
3V
Finally let r iy and <$*, factor of
in
&
the highest common as both considered polynomials in y, z, the domain of ratioThis highest common factor is obtained by i nality being
i
i
.
.
y
.
9
r be
.
the Euclid algorithm, bearing in mind that a polynomial ? u q is zero when and only when the polynomial c, in 1 MI in Wj obtained by replacing 1 by w, in ?, is the product ,
A
by
,
,
w
of a polynomial in
We
with coefficients in
.
shall prove that the highest common factors just found Let t 0, the irreducible factors of a in become, for
=
.
a be a resolution of
A where each
ipi
=
Vi
results from
Vk
V>2
y>i
each ^i>
It is easy to see that
irreducible in <3)1
Then
into irreducible factors.
on replacing y by y as a polynomial in
z^.
t/,
z, is
of
some
.
^
Manifestly each
is
common
a
factor of
we can
A
and
^
the case, prove that, our result. and (5;, we will have highest common factor of Let #^, for^" m, be the polynomial obtained 2, from V on replacing 1 by j. Let dj in
(18).
If
in this
is
A
,
=
^
(19)
Then
171
is
a polynomial in r
Each *
^^^
di in (18) is
Perron, Algebra,
=
a factor
t/,
vol. 1, p. 210.
.
^ with coefficients in 3)
?i ?*
of
m)
?fc.
some
iy;,
?
and
ALGEBRAIC DIFFERENTIAL EQUATIONS
76
Suppose, that Vi of
f]i.
of
fli
a factor of d 1 and that d l
is
we prove that ty is and //! we shall have our If
^
Suppose, for instance, that
a factor
is
common
the highest
{
factor
result. is divisible
by
^
i/> 2
Then,
.
by (19) (20)
v;'---
where
The
a polynomial in
is
^
with coefficients in
0,
?/,
set of terms of highest degree in
of (20) is of the
T/,
z in the first
m
and
j.
member
form
(21)
with & a rational combination of the
The terms
/
second member give an expression
in the
of highest degree of the type
*(#,*)
(22)
Now i>l
(y-*W.
cannot be equal, since no y #& with is divisible by y z^. This completes the proof. 51. We consider a non-trivial prime system 2 in the un-
and
(21)
knowns
Mi,
i/i,
(22)
for
which AI, Av,
(23) is
a basic
,
J^
each Ai introducing yt. In 53, 54 we show the At are given, a simple resolvent can be con-
set,
how, when
structed for 2.
52.
Let A
represent
t
,
2 when
,
Ap be
the
unknowns are the m, A,
show, as in 17, that no non-zero form in the to
We 2
of
t
^ ?(i,
is
indecomposable.
k
easy
holds 2\.
with the same m, k, for which
Q
=
^i2/iH
-----
G
\-ipyp
gives two distinct functions of x.*
At
It is
Furthermore,
43 (or 25), that there exists a non-zero in the m, ^ such that, for two distinct solutions
the form
*
i/i.
see as in
O
form
Let 2A be used to
new unknowns.
present,
we have no way
of determining
G.
does not vanish,
ALGEBRAIC SYSTEMS
IV.
44, a simple resolvent exists for
By
&
Let
2
t
,
77 for
which
w
= Q.
be the system of all simple forms in the U{, h, w, yi 21 and w Q. We consider a basic set for &,
which hold
R,
(24)
which which B
in
iv y yi,
B l9 -",BP
yp are introduced
,
algebraically solvent for 2\ and each Bi
We
53-
in
shall
and
succession
Then
irreducible.
is
B=
is
in
a re-
linear in yi.
is
now show how
a basic set (24) can actually
be constructed. the method of elimination of
By by a
finite
form 8
number
w, the and w (23)
of
^
and
Q
for
in
Then 8 belongs
31,
we can
determine,
of rational operations, a non-zero simple
&.
to
iii which vanishes for all solutions which no initial in (23) vanishes.
Now, o ^^
let
op
&i
Then with each Si algebraically irreducible relative to The selection of such an Si can be made some Si holds .
.
as follows.
Consider any Si and
let
T
be the form obtained
For Si to hold ., it is on replacing iv by Q. T Let T be arranged hold 2 t necessary and sufficient that For Tto hold 2l9 it is necessary as a polynomial in the A t-. from
it
.
and
A
each coefficient in the polynomial hold 2. 2 if and only if its remainder with
sufficient that
coefficient will hold
respect to (23)
is
zero.
Every form in the in, ^ and w which holds & is divisible by B. Thus an irreducible factor of 8 which holds SI must
B
be the product of in (24) by a function in 3). We have then a method for constructing a simple resolvent for
2lt
It
remains to show
how
a complete basic set (24)
can be determined.
Let
U
be the form which results from
by
w + yi
of
U
in yi
and A x by is
highest power
Ai
that
of-
of
in
y
+ 1. B U
in is
Then
w
U
and the
free of
B
holds
ic.
on replacing .
w
The degree
coefficient
of
the
ALGEBRAIC DIFFERENTIAL EQUATIONS
78
Now
represent any analytic function of the which makes vanish when substituted for w. let
R and x a be the polynomial in Let
y^ obtained
=
cc
(25)
(*!
as
on replacing <*
w
in
HI, Ii
Let
?7by f
.
m
be a decomposition of a into irreducible factors obtained as The coefficients in the a* are rational combinations 50.
in
of
m
the
,
and
Let
A$.
of these coefficients.
The r are in
be the product of the denominators
irreducible and their coefficients are polynomials
the M* and ^.
,
/tf
Then
by w.
on replacing
Let B be the form which results from ft Let Ci result similarly from y<. Then
BUCi
(26)
Cm
vanishes identically in y t when w is replaced by as in hence that some 46, that (26) holds ,
Suppose that Ci that Ci
linear in
is
is
found (by
ylf
If
is
/!
d
holds
hold &.
to
test)
It follows,
.
We
.
say
the initial of RI in (24)
we have (27)
where
R. is
K
ylB As isT holds /2, it is divisible by Ci were not linear, (27) would imply that Y\
is free
Thus,
if
of
reducible.*
only necessary, then, to take the remainder of Ci in order to have a form which will serve
It is
with respect to
R
as
l
R
in (24).
i>l
The Ri with
in
(24)
are determined in the
same
way, 54. It remains
Ii
Let and
Jlf,-,
*
I
be the
now
to construct a resolvent for 2.
initial of jR, in (24)
and
R
We
note that Ii cannot vanish
when w
is
Ri. As can find forms
Ii that of
are relatively prime polynomials, Nij Li such that
we
replaced by
C.
ALGEBRAIC SYSTEMS
IV.
+ NiR =
Mili
(28)
Li
with Li free of w, and not zero. Let the A< be replaced by integers JL J-i
We
does not vanish.
2
Let
For the
a yl l
its irreducible
for these
put
which
factors in
h is
=
Let
(Z>.
One
.
R
r
be resolved into
factor $,
which can be
a*
with respect to 8.
we
see that no R'i has an Let Si be the remainder of Then each form in the set
in
divisible
(30)
a t-, (24) becomes a system
R{, ..., 5;.
of (29) holds
determined, will hold
we
=
-
#,
Then each form
R'i
(24) gives,
with
----- ap yp
substitution A t
(29)
If
that
----aiyi H \~0pyp*
w
initial
way
be the indecomposable system obtained by adjoining
to 2.
such
Lp
show how
shall
=
w
en in
J.p JL/I
substitutions, a resolvent for
79
(28),
by S.
&,
8,
,
Sp
holds 0. It
is
clear
that (30)
a basic set for the totality of is a To show that S
is
simple forms which hold
=
CD.
simple resolvent for 2, we have to prove the existence of the form G of 43. If two distinct solutions of CD have the same
Ii
of
m
some
and w, the
m
and
Si in (30) vanish,
prime polynomials,
we have
w
Aus
must make the initial I'i and 8 are relatively
a relation
MiH + NiS
=
Li
with Li a non-zero form in the m alone. Then ii which can actually be constructed, will serve as G.
Lp
,
ALGEBRAIC DIFFERENTIAL EQUATIONS
80
RESOLUTION OF A FINITE SYSTEM INTO INDECOMPOSABLE SYSTEMS 55. Let
not
2
be any finite system of simple forms in y v --,yn In this section, we show how to determine ,
zero.
all
of a finite number of prime systems which form a set of systems equivalent to 2. Later, we shall obtain finite systems of forms equivalent to the prime systems.
basic sets
Let
^,^,..-,4,
(31)
be a basic set of 2, determined as in 4. If A is of class zero, 2 has no solutions and is thus indecomposable. We assume
now
that AI
not
is
class
of
For every form
zero.
in
2,
the remainder with respect to (31) be determined. these remainders are adjoined to 2, we get a system
If
let
4, if not all remainders By equivalent to 2. 2' will have a basic set of lower rank than (31).
by
that after a
3,
finite
number
2''
are zero,
We
of repetitions of the
see,
above
operation, we arrive at a finite system A, equivalent to 2, with a basic set (31) for which either AI is of class zero
or for which otherwise is
remainder of every form in
the
A
zero.
Let us suppose that we are
make a temporary set (31) for A, A n by yr The q
=
relettering of class
is
i
in the latter case.
the
of
j
we
t ,
y^
If,
We
shall
in the basic
replace the symbol
p unknowns not among the
y.
we
^
call,
uq We list all the unknowns in the any order, u i9 u order u v yv yp q With this change of notation, we proceed to determine, 45, whether (31) is a basic set for a prime system. using
in
-
.
,
If
A
where
l
is
J9 X
=B
reducible, as a polynomial in y1 and if A l are of positive degree in y v then and 2
B
be equivalent to A-\-Bi, systems, after will
.
,
;
,
we
A + B*
then
analytic function
of
l
jB
2
,
will
the latter
revert to the old notation for the unknowns,
have a basic set lower than
Suppose
and each
A
of
substituted for y r
(31).
and
'
be any Ui,-'-,u q which annuls AI when Let <* 2 be the polynomial in y2 which
that AI x\
is
irreducible
let
ALGEBRAIC SYSTEMS
IV.
AS becomes, for
G
Suppose that
this substitution.
as a polynomial in
?/ 2
2
reducible
is
(10), there exist non-zero
By
.
81
forms
reduced with respect to AI A s such that /i (?i 6^ holds ^. Of course, 50 furnishes a method for actually and O Then is equivalent to 2 determining GI Gi
,
2
,
,
^
.
Each
of the latter systems, after
for the
we
unknowns, has a basic set
This becomes clear
,
revert to the old notation of
lower rank than
(31).
K
one considers that, in (10), T, Ot, G*, do not involve any ui not effectively present in AI and J 2 Suppose now that 2 is irreducible. Let A and A 2 vanish if
.
for y^
from
=
A8
',
2/2
If
.
from (13) that
=
#3
"
Let
is
the usual manner
in
t/3
we
,
see
equivalent to
each of which latter systems, basic set than (31). for resolving 3 into
result
with respect to
reducible
is
A
8
its
has a lower
in the old notation,
What we
need, however,
The
irreducible factors.
a
is
method
irreducibility
A 2 is a basic set of 2 show that AI properties of A l and 1 -= Let be a prime system in u iy u y v y2 a simple resolvent for A' constructed as in 54, with ,
A
-
,
E
.
;
,
w
(32)
#i,
a2 being
when in
a l yl
a^y2
It is clear that
integers.
substituted for w, and that
aiC'
+ ait",
any form
member
C'
and
then
C vanishes
(7
if
0,
ai'
+a
" 2
annuls
E
and f" are each rational
with coefficients in
in the w<, yi
of (32),
=
.
In short,
if
is
?^, which holds A' and the first results from B on replacing w by
and
C
holds A'.
Then
=
". Thus, in factoring 8 y == f y2 we may use the domain of rationality obtained by adjoining
so that
for
;
,
,
ALGEBRAIC DIFFERENTIAL EQUATIONS
82
and
the \a
fliC'
plished as in All in
A
The
to 3).
factorization
is
accom-
A
we have
all,
whether (31) replacing
+ OgC"
50.
a method for testing to determine set of a prime system, and for
basic
a
is
set of systems each with basic sets lower
by a
than (31) when the test
is
negative.* notation for the unknowns, let us suppose that (31) ha,s been found to be a basic set for a prime is equisystem. Let 2t denote the latter system. Then valent to
Using now the
old
A
Each
A + Ii
What
has a basic set which
lower than
is
(31).
2
precedes shows that the given system
can be
resolved into prime systems, as far as the determination of basic sets of the prime systems goes, by a finite number of rational
and factorizations,
operations
same can be done
for
all
provided
that
the
systems whose The final remark of
ibasic
sets
finite
are lower than those of 2.
3 gives
a quick abstract proof that the resolution is possible for 2. What is more, the processes used above, of reduction, factorization and isolation of prime systems 2l give an algorithm ,
for the resolution.
56. It
remains
to
the
solve
problem:
following
Given
a basic set
4i,---,4,
(33)
of a non-trivial prime system of
class q
+ (p + q = i,
ri),
in yi, it
system of forms equivalent to 57- Let (34)
*If,
Zi
when
the
=
unknowns
in yi
are
+
is
-,
yn
required
each Ai being
,
to find a finite
.t
Mm
----
Ui,
uq
,
\
y
l
,
=
i
2/n,
,
yp
,
1
,
,
n,
(31) is a basic
the prime system 1*, then, when we revert to the old notation, (31) will he a basic set for the system into which 12 goes. set
1
for
2
of
unknowns
55 leads to several systems
the various 12,
we
&
.
For each
12
,
After finite systems are found, can revert to the original lettering.
appropriately.
we
reletter the
equivalent to
ALGEBRAIC SYSTEMS
IV.
where the the
Zi,
we
find,
Zi,
new unknowns.
are
tij
83
Given any g
+1
of
by the method of 31, a non-zero form in those z\ and which vanishes for arbitrary ty, provided that the Zi are t\j obtained, according to (34), from yi which satisfy (33) and make no initial in (33) zero. Let m be the Let B be such a form in 2i, 2^+1. the
,
B
degree of
considered as a polynomial in the to obtain a relation C ==
show how where
C
among
m in
addition, is of degree
Let (35)
m
of degree
is
Zi
=
as a polynomial
each
Zi
We
shall
,
z q +\,
Zi,
the
in
separately,
Zi.
in
and,
z%
i=l,
a,i*iH-----ha*,
and the ay are new unknowns. 0, B' being goes over into a relation B' a polynomial in the z\ whose coefficients are simple forms in the ty, ay. The degree of B' in each z\ will be effectively m.* Furthermore, we can specialize the ay as integers, in such
where the
z\
=
B
Then
and that the becomes a nonLet this be done, and let B" zero simple form in the ty. be the form in the zt, ty into which B' thus goes. a
way
that the determinant
coefficient of the
From
mth power
(34), (35),
we
is
ay\
of
each
not zero z\
in
B
r
find
(36)
z(
=
where each T^
is
a linear combination, with rational numerical
rnyi-\
-----h*tny,
coefficients, of the ty with
see that the
ty
i<, q
i^q+1
with
+1
.
*
=
From
1,
(35),
,
2+1,
(36),
we
are linear combinations of
the Ty, with integral coefficients.
Hence, the Ty
may be
made to become arbitrarily assigned analytic functions, the ty are taken appropriately. In the relation B" 0, we substitute, for each ty,
=
expression in terms *
of the Ty.
Perron, Algebra, vol.
1
,
p. 288.
Then B" goes over
into
if
its
a
ALGEBRAIC DIFFERENTIAL EQUATIONS
84 form B'"
B
in the
g'l9
=
i
Ty,
1,
,
We
g+1.
now
replace, '"
u>
Then each T^ by fy- and each zj by z%. goes over into a form C in *i, ^1-1 and the ^/, (7 being of degree w as a polynomial in the z\, and of degree m in
in
',
,
Furthermore C vanishes for all zi given separately. for the which by (34) yi satisfy (33) and make no initial zero. This is because (36) may be considered equivalent to each
the
*j
first
+l
q
relations (34).
(7=0
Evidently the relation
will
of the z^
zq
we replace that a correprovided subsist
\.\ by any
z\,
,
We
now
specialize the
if
(7.
j in (34) as integers
tl
with a non-
vanishing determinant, in such a way that the relations obtained from C for the various sets of g+1 unknowns
remain of effective degree m in each zi appearing in them. These relations will have coefficients in 5).
We
with the ty fixed as above. If the y are replaced in (33) in terms of the z^ we get a system & of p forms in the Zi. Let basic sets be deterLet mined for a set of prime systems equivalent to CD. 58.
consider
z\,
,
zn
t
2 i9
2S
,
be those prime systems which are not held by
the initial of any At in (33), the yi being replaced in the There will be one of the 2i which initials in terms of the **.*
holds
the
2
remaining
This
Z .
is
in
because,
a resolution
indecomposable systems none of which holds is precisely one which is held by no initial. there any other, To determine that - which holds the others, all we need of
into
(33)
t
do
to
is
Suppose,
22
,
for
2a
..-,
a
find
some form
^
if
2i
is
*
obtained from
Zi.
We
2
l
the basic
in
surely 2j cannot hold no 2j can hold all 2t of the
that
instance,
Then,
.
whose basic
holds
basic
the
set
2lt .
/2,
of
other
^
2iu
holds
does not hold 2j, the initial of set of 2 l must hold 2j. Then
Thus if 2 does not hold Then holds all 2if *
all
J
t-,
^
(
56) by replacing the yi in terms
shall prove that
The condition
the
set
2l9
for a form to hold a
with respect to the basic set vanish.
like fi,
has q unconditioned
prime system
is
that its remainder
ALGEBRAIC SYSTEMS
IV.
unknowns
#+1
forms in any
On
To begin
42).
(
2
in
1
,
if
we
a non-zero form in
found in 57, belong to 1\. there were fewer than q unconditioned could use the basic set of ^i to determine of the z t-,
the other hand,
unknowns
easy to see that the
is
it
with,
85
yi,
,
y q belonging to
-Q.
z q be unChanging the notation if necessary, let zi, conditioned unknowns for 2 lt Then 2 will have a basic ,
l
set
S
(37) in
BI
which BI introduces
...,BP
l9
We
z q +i.
assume, as
algebraically irreducible. construct a simple resolvent 59-
we may,
that
is
R=
We
w
(38)
=
-----
rtiZ0.t-H
the ai being integers. Let We shall prove that the
R
be
for
^
,
with
\-aptq-i-p9
of
initial
degree g in iv. is a function of x
of
R
According to 57, each z, ?>g, satisfies with an of 1, equation degree w in zj the coefficient of zf 2V/ of a x in function We may and shall assume that 5). being the coefficient of z? i>#, in each of these p equations is in
.
?
?
1
,
Then
unity.
(38)
shows that
w
satisfies
with
zi,
,
zq
an
equation in which the coefficient of the highest power of w is unity.* This implies that, in the algebraically irreducible simple form
R,
the coefficient of
We may and shall assume that the We shall show that Eio + Emv+. -/QQ = ------Zi (39) x
i
-
= g-f
1
.
?
.
., 71,
Mzq +i where
Jtf
than g in we have
is
N
is
free of zi,
coefficient of
wg
is
,
zg
.
unity.
-
where the Ey and
Let
wg
N=
D are forms in zi,
-,
zq
.
0,
and are forms in w\ zi, zg of degree less w. As jlf and J2 are relatively prime polynomials, ,
* This is analogous to the fact that the sum of several algebraic integers an algebraic integer. See Landau, Zahlentheorie, vol. 3, p. 71.
ALGEBRAIC DIFFERENTIAL EQUATIONS
86
PM+QR = L
where
a non-zero form in
is
Lzq+
zi
L
y
zq
,
iPN =
Then
.
0,
PN
and, replacing by its remainder with respect to R, we have a relation (39) for z q +i. Evidently we may use the
same
I) for
Let
60.
^+1, MI,
,
,
zn
<
be new unknowns and
v
Up\
the totality of simple forms in the
2
and
i
V
ut
gi,
- UiZq+i ----- Up g
n
-A be
let
and v which hold
*
A
has an algebraically irreducible form Zin v, zi, '-,z q and the m, the coefficient of whose highest power of v is
Then
unity.*
We we
Z
shall prove that
is
of degree
in v.
g
Using
(39),
see that
where the K* are simple forms in Zi, z q and the w*, and For the first where w is given by (38). g powers of v, we ,
R
get similar expressions if we make use of fer, by a linear dependence argument, that v z\j
zq
,
most g
and the
Ui,
if
Z)^0, an
The condition that i> that A is prime. Thus Z is
Z
w
We
0.
satisfies,
in-
with
equation of degree at is removed by con4"
in v.
sidering On the other hand, as v becomes
=
at
if
most
w*
=
of degree
a/,
i
g
in v.
= ,,#, 1
cannot be of degree less than # in v.t Let v be replaced in Z by
Mi*f+iH-----I-WP**.
(40)
^ becomes a form in *n and the w. Let this form be arranged as a polynomial in the m with coefficients which are forms in the 2*
*!,,
Then
5
,
* Note that each Ui zq +t satisfies an equation with the coefficient of the highest power of M
Z
=
ALGEBRAIC SYSTEMS
IV.
^
Let the
W
be the
finite system of these coefficients (forms in are going to prove, in the following sections, equivalent to 2lt Thus, if the Zi are replaced in
We
Zi).
^
that
is
we
their expressions (34),
by
We
equivalent to .. 56. stated in
We
61.
z
sufficient
as just
ss
q
Q
thus have solved the problem
shall
Z
given,
^
W
of
D
w
&
-
W.
,
,
q
,
But v
.
g
we
,
%q
,
solutions for w.
,
the
Using
get g distinct solutions
*o') ,
.
^4-1
.
.
>
>
*O) ^ n
i1
?*
>
J
-
.
.
/T y
.
>
>
Z becomes
for
Zi
,-,
Then
up %n
MiJg+i
that for some j, our statement.
We
distinct
be the polynomial which
>J -
,
Corresponding to ?1?
gives g
in (39), ...
Let
1
DO, .
=
si,
=
let
We
B
of ^i.
This
w and
with respect to
K^
equation each such
and for
(40),
as in (39). shall prove that every solution Let 5X is a solution of 2^ n be
is
such a solution of
i
R
be the discriminant of
with
in
vanish for arbitrary m. holds 2.
K= where
as
v
for
that,
shows, in particular, that
Let
system of forms
finite
,
,
and
necessary ,
get a
begin with the observation that for given analytic z n to constitute a solution of *F, it is
functions
*i,
87
gfJ>
=
&,
is i
a factor of
=
-
q-}- 1,
-,
This shows
ft.
n, and proves
show that, given any solution 1, being analytic in some open region 2li in 21, in which there exists an open region 21', contained in 2l t 62.
of
W, the
are going to
,
?i
,
the solution can be approximated uniformly by a solution of for which, throughout there exists a solution
such that IT out
21',
f
=
+ 1,
21',
^
,
K^O. ,
^ of
That
|
&
^<
|
<*
n. 7
W
*>0,
V, analytic throughout
throughout 2f and that ,
is,
for every
21',
through-
ALGEBRAIC DIFFERENTIAL EQUATIONS
88
This will show that 2i holds f, for since a form in 2 vanishes for every solution of *P with 0, it will vanish, shall thus know by continuity, for every solution of W.
K^
We
2
that *P and
We
63.
are equivalent. shall establish the more general result that
if
H
,
any non-zero
,
W
of
W It
H distinct
with will
which
We
,
from zero throughout
evidently
assume
2l x
3+1,
,
to zn
their coefficients
and the
There
H
is
Ey
-
Let
61
,
throughout
analytic
H,
K
in
21 A
of
We in R
.
61,
assume also 59 and of
of (39) are analytic throughout 2l x
,
,
.
We make this assumption. be constants such that
by K. bq
does not vanish for every x.
Then,
if
h
is
a complex variable,
,, + M)
#(?1+M,
(41) is
-
evidently no loss of generality in assuming that
divisible
is
for
be so taken that the equations of degree m zq ( 59) have each satisfy with gi,
that the coefficients in
D
W
H=0.
which
in
21'.
consider a solution of
to
suffice
a polynomial in h of the type
a r hr -\-----\-<* s h 8
(42)
where the vanishes
are functions of
<
for h = 0,
we have
x
,
Since (42) assume that a r is
analytic in 2li.
r 2> 1.
We
not zero for every x. Let 212 be an open region in 2li in which a r is bounded away from zero. Let h be small, but distinct from zero.
Then
(42) cannot be zero at *
(43) jg
=o
in S1 2
.
will
This
any point
= & + M,
of 21 2
Thus,
.
*
=
1,
if
,,
have g distinct solutions for w, each analytic is divisible by the discriminant of E. because
is
H
ALGEBRAIC SYSTEMS
IV.
H
As
divisible
is
D
by
in
89
(39), [Si will
have g distinct
solutions with Zi, zq as in (43), for which z q +\ 9 are given by (39) and are analytic throughout 21 2 ,
,
zn
.
Consider a sequence of non-zero values of h which tend
towards zero,
hi,hi,-..,Jn,-..
(44)
each out
being so small that (42)
hi
21 2
For
.
eacli
(45)
Z
zj
will vanish
if
(46 ) A;
if
7^,
i;
=1,
=
Ui
=
distinct
from zero through^
Zj+l)jhi,
^i+
where the zf
,#,
is
...
are
}
+Up
=
1,
,?,
^ 21
analytic throughout
2
.
depend on 7i r For distinct are from one another. the any hi, g expressions (46) As the equation of degree m which each Zj, j>q satisfies
It is understood, of course, that the
with z x there that,
-
,
is
sf
has unity for the coefficient of zj1 ( 59), a region 21 8 in 2I 2 and a positive number d such -,
z
,
q
21 3
throughout
,
(47)
zf
= q+
|
=
n\ k 1, # and for every In in (44). " This is because the coefficients of zj1 1 -, gfj, in the above considered equation, are bounded quantities. for j
1,
-,
,
,
For each
In of (44),
let
one of the g expressions (46) be We form thus a sequence
(i> selected, and designated by v
(48)
t/,
Let
.
*", ---,
!/, -...
be any bounded open region which
21'
lies,
with
its
From (47) we see, using a well known boundary, in 21 8 theorem on bounded sequences of analytic functions,* that, .
some subsequence
for
i=l
... y
function. of
m *
be
j
of
(48),
the coefficients of each m,
p ? converge uniformly throughout Let the limit, for the subsequence, {.
We
find thus that
Montel, Les families normales Dienes, The Taylor Series, p. 160.
21'
to
an analytic
of the coefficient
if
de -fonctions
analytiques,
7*
p.
21.
ALGEBRAIC DIFFERENTIAL EQUATIONS
90 (49)
Z
=
*,
&,
j
=
l,
vanishes for
Deleting elements of (44) if necessary, we assume that the convergence occurs when the complete sequence (48) is used, rather than one of its subsequences. For each hi there are
g
not used in (48).
1
expressions
1
expressions be selected for each
(46)
Let one of these In, and let (48) be
g used now to represent the sequence thus obtained.
we
select
As above, a subsequence of (48) for which the coefficients
m
of each
converge uniformly in
expression V
which causes find
Z
Ml
to vanish
g expressions
(50)
=
=
=
t;
This gives a second
21'.
l
when
(49) holds.
+...+t pC5f Z vanish when
CS. 1
we
Continuing,
)
f
>
1, ., g, which make (49) holds. Let vk represent the second member of (50). Again, let Wk represent the second member of (46), it being understood
k
.
that the subscripts k are now assigned, for each hi, in such a way that the coefficient of in in Wk converges to that in Vk as hi
Then
approaches 0.
since the
g expressions Wk are
we
other for every h i9
Z
polynomial which
By
continuity,
1
,
,
Jn V
must be a factor
wi)
-
represent Z,
=
Y
But since
(v
we
if
have,
becomes when
=
ft
will
(v
is
from one an-
representing by
ft
the
(45) holds,
(v
wg )
when
.
(49) holds,
by
vi)-- (vvg).
a solution of
Ui $ q +i
of y.
-
distinct
*F,
----- Up %n
This shows that, for some &,
y,
ALGEBRAIC SYSTEMS
IV.
91
This establishes the result stated at the head of the present section and proves that *P is equivalent to 2lt
A
We
64.
THEOREM: Let in y
,
,
SPECIAL THEOREM
prove the following theorem.
yn
2 be
Let
an indecomposable system of simple forms
B be any simple form which
does not hold 2.
Given any solution of 2, analytic in an open region 2li, there is an open region 21', contained in 2li, in which the given solution can be approximated uniformly, with arbitrary closeness, by 2 for which B is distinct from throughout 21'.
solutions of
We
assume, without loss of generality that 2 is prime. 57 is effected, 2 may be replaced the transformation of zn by JSi, while goes over into a form C in ^, be C does not hold -Si. Let g-H, ># replaced in C If
B
their
by
of ^i
W
,
,
expressions
with
D^
(39).
;
find
for
that,
all
solutions
,
ri ^
fxiA \
We
.
^ '
~f)t*
N
In (51), w is a simple form in w] Zi, zq member of be the second (38). Because supposed given by of 59. Thus, we have does not hold J5i, Ni& not divisible by
where
is
-
.
,
DC
to
R
XE+YN =
(52)
where Let
H
that
if ,
a non-zero simple form in
Zi,
-
-
-,
zq
.
be a region, contained in 2li, in which the coefof the forms in (51) and (52) are analytic. We see a solution of J5i makes C vanish at some point c
21 2
ficients
in 21 2
is
H,
then
DH vanishes
at c.
which any given solution uniformly by a solution for which from zero throughout 21'. As the the Zi, we have our theorem.* in 21 2 in
But there is a region 21' of J5i can be approximated DH, hence C, is distinct yi
vary continuously with
*This useful theorem, and the considerations which lead up to it, do not seem to exist in the literature, even for the case of equations with constant coefficients. Professor van der Waerden recently communicated to
me
a different proof, which deals with the case of constant coefficients.
CHAPTER V CONSTRUCTIVE METHODS CHARACTERIZATION OF BASIC SETS OF IRREDUCIBLE SYSTEMS 65. Let
AI, A*,
(1)
-
Ap
-
-,
be an ascending set of differential forms in UL,
,
Uqi
y\_
+
We
i. each Ai being of class q and sufficient condition for
,
,
yp
,
are going to find a necessary be a basic set of a closed
(1) to
irreducible system.
Let the order
i
=
1
.
9
.
.
7
p.
of At in yi
We
be n.
The remaining
in (1)
ytj
represent y^ by Zi, and the ny present
we designate now by symbols vie, attributing the subk in any arbitrary manner. With these replacements of symbols, (1) goes over into an ascending set of simple
in (1),
scripts
forms,
B19 B
(2)
in the
9
JBP
-,
,
unknowns Vr
(3)
V
The passage from
(1) to (2)
,
,
'
1,
\
'
'
,
Zp*
purely formal. Once it is effected, we treat (2) like any other set of simple forms in the vi, Zi. For instance, whereas, in a solution of (1), yij+\ is
must be the derivative of i/
.
92
CONSTRUCTIVE METHODS
V.
We is
prove
not
first
fulfilled.
see that,
since
the necessity.
93
Suppose that the condition 55, and also 46, 47, we
Referring to is not a basic set of a prime system,
(2)
there exists, for some j, an identity
jft
.
.
jfr (TB.
.
where
Ii is the initial of
the unknowns in to
&W -K.B, ----- K^Bj,, =
JBi,
#!,...,,. To this identity,
the
m,
(4)
Jp-J
0,
Bt and
,
there corresponds an identity in forms in
yi,
H
H
and Here 2 are non-zero with reduced which are respect- to q-}~j, a closed irif a basic set of were AI, (1) Thus, Aj. reducible system, either jff2 or some Ji would belong to
where
J* is the initial of Ai.
forms
of
class
,
H
the system.
,
This completes the necessity proof.
Suppose now that the condition is fulfilled. We shall prove that (1) has regular solutions. Consider any regular
first
solution of (2).
Let a be a value of x for which the funcand the coefficients in (2), are analytic,
tions in the solution,
and for which no initial or separant in (2) is zero (for the given solution). Let the values of the Vi, z* at a be assigned to the
corresponding w#, yg in
We
(1).
construct functions
M!, Uqj analytic at a, for which the w
derivatives of yi, at a, the numerical values associated with them above, the differential equation t will have a
A
=
regular solution in which the Ui are the functions above and which the first n derivatives of t/i have, at a, the above
in
We
uq y into A% and substitute MI, , assigned values. for the initial conditions determined above. solve y2 with \
We
have now a regular solution
find
a regular solution of (1). let G and be two forms such that
Now, for
all
of
AI At ,
H
regular solutions of (1).
Let
(?x
.
Continuing,
OH
we
vanishes
be the remainder
ALGEBRAIC DIFFERENTIAL EQUATIONS
94
O
with respect to (1), and HI the remainder of H. Then GI HI vanishes for all regular solutions of (1). It may be that GI and HI involve certain mj not effectively present of
in (1).
In that case,
for the
new
let
Then
M#.
system even after
new symbols
(2) will
vt
be added to
(3)
be a basic set for a prime
this adjunction
of
continue to satisfy the condition of
unknowns, for
will
it
45.
As we saw above,
the values of the functions in a regular a point a which is quite arbitrary, are This values of the uij, yij in a regular solution of (1). from are obtained and if G and GI 2 HI by 2 means, 2 vanishes for replacing the Uij, yg by the vt, zi, that G2 Hence either G2 vanishes for all regular solutions of (2). solution
at
of (2),
H
H
all
regular solutions of (2) or
H
2
does.
Suppose that
G2
does.
As G2 is reduced with respect to (2), G2 vanishes identically. Then G vanishes for every regular solution of (1). Thus, the totality solutions
of
shows that
(1) if
is
2
of forms
which vanish for the regular
What precedes belongs to 2, the remainder of G This means that 2 has no nonzero.
an irreducible system.
G
a form
with respect to (1) is zero form reduced with respect to (1), so that The sufficiency proof is completed. set of 2. 66.
We
shall prove that if (1)
irreducible system
is
a basic
2, then every solution of
separant vanishes is a solution of 2. Let Si be the separant of At. Let vanishes for all regular solutions of (1).
show the existence
of
integers
$1
,
,
G
of a closed for which no
be a form which in
5,
=
O l9
is
by
,
Ap
not of higher order than any Ai in
!,..., p. Let result from GI when
Then
we can
such that, when
a suitable linear combination of derivatives of AI, with forms for coefficients, is subtracted from
the remainder,
a basic
set
(I)
As sp
(1) is
,
yi,
H we pass to the unknowns H vanishes for every regular solution of Hence, H for vanishes solution of for which no 49, every (3).
(2).
(2)
CONSTRUCTIVE METHODS
V.
95
separant vanishes. Then GI vanishes for every solution of So does (7. This proves (1) for which no separant vanishes. our statement.
BASIC SETS IN A RESOLUTION OF A FINITE SYSTEM INTO IRREDUCIBLE SYSTEMS
67. Let
not
2
zero.
all
be any finite system of forms in yi yn, In this section, we show how to determine
basic sets of a finite
which form a
,
7
number
closed irreducible systems In Chapter VII,
of
set of systems equivalent to 2.
we
give a theoretical process for determining equivalent to the closed irreducible systems.
finite
systems
Let AI,
(5)
A 2j
-
Ap
-
-,
If A ly 4. be a basic set of 2, determined as in class zero, ^ has no solutions, and is thus irreducible.
is
of
We
assume now that AI is not of class zero. For every form in 2, let the remainder with respect to (5) be determined. remainders are adjoined to 2, we get a system 2' 4, if not all remainders are zero, equivalent to 2. By 2' will have a basic set of lower rank than (4). We see,
If these
by
that after a finite
3,
number
above
of repetitions of the
operation, we arrive at a finite system A, equivalent to 5, with a basic set (5) for which either AI is of class zero or for which, otherwise, the remainder of every form in Let us suppose that we are in the latter case.
A
is
We
zero.
shall
make a temporary relettering of the yi. If, in the set (5) for Ay At is of class ji, we replace the symbol yj by yi. The q n p unknowns not among the yj we call, in
=
any
order,
t
t ,
MI,
order ti,
With whether
,
this (5)
If it is not,
Each will
,
uq
uq
We
.
all
list
yi, yp change of notation, is
we
\
,
a basic
(4),
we
that
of the latter systems, after set
unknowns
in
the
determine,
by
65,
set of a closed irreducible system.
see from
have a basic
the
.
we
lower than
A
is
equivalent to
revert to the old notation,
(5).
ALGEBRAIC DIFFERENTIAL EQUATIONS
96
when
If
the
unknowns
of a closed irreducible
the
old
notation,
(5)
are the
m,
yt,
(5) is
a basic set
system ., then, when we revert to will be a basic set for the closed
irreducible system into which Q goes. Using now the old notation for the unknowns, let us suppose
that (5)
has been found to
irreducible system.
by
66,
A
sl
+ Si
Each
What
is
Let
^
be a basic set for a closed
Then.
denote the latter system.
equivalent to
has a basic set which
is
lower than
precedes shows that the given system
(5).
2
can be
resolved into irreducible systems, as far as the determination of basic sets of the irreducible systems goes, by a finite number operations, differentiations and factorizations, that the same can be done for all finite systems provided whose basic sets are lower than those of 2. The final remark rational
of
3 gives an abstract proof that the resolution is possible What is more, the processes used above give an algorithm for the resolution. of
for 2.
In the resolution into irreducible systems obtained above,
some systems may be held by
others.
The algorithm obtained above contains in itself a complete elimination theory for systems of algebraic differential equations. get all of the solutions of 2 by finding the solutions of
We
each basic set which cause no separant to vanish. A solution of an irreducible system which annuls some separant will be a solution of some system like A -f fli above, and hence will ultimately be found among the solutions of some other Thus our irreducible system, where it annuls no separant. of the reduces of solutions all process algorithm determining
a system of algebraic differential equations to an application of the implicit function
theorem and of the existence theorem
for differential equations. It
yi>
what precedes that a system of forms in in which each form is linear in the yij, is an
follows from >
l/ny
irreducible system.
CONSTRUCTIVE METHODS
V.
97
TEST FOR A FORM TO HOLD A FINITE SYSTEM 68.
Let
be any finite system of forms. Let it be determine whether a form holds required given
to
.
The
hold each irreducible system.
that
condition for
G
to hold one of the irreducible systems is that its remainder
with respect to the basic set of the irreducible system be zero. This gives a test which involves a finite number of steps.
CONSTRUCTION OF RESOLVENTS 69.
Let
A2
A\,
(6)
where the At are forms
in
,
Ap
,
wu
uq
,
,
y
\
-
-,
,
yp
,
each Ai of
be given as a basic set of a closed irreducible suppose that either ^ does not consist entirely
class q-\-i,
We
system J.
of constants, or Ui actually exist.
We We
show how
shall
to construct a resolvent for 2.
begin by showing how to obtain the form G of Let Bi be the form obtained from At, by replacing each
new unknown
We
consider the finite system of of the forms (6), the forms posed
a
zj.
B19
(7)
25.
by com-
yj
--.,BP
and also
*iH
^1(2/1
where the Ui
A* '
j
'
are unknowns. '
>
Uq
;
/!
-
,
-
l-ipfyptp),
We order the unknowns as follows: /,
,
p
;
y
,
,
yp
;
z\
,
,
Zp .
67 for resolving Q into irreducible apply the process of each irreducible system being represented by a basic systems, set. The theory of 25, 26 shows that each irreducible
We
not held by every form yi Zi has a basic set containing a form in the Ui and A alone. We obtain, of 25. by a multiplication of such forms, the form
system which
is
K
When & contains of
(AI
a non-constant function, the determination which do not annul L of 25 is an elementary problem
ALGEBRAIC DIFFERENTIAL EQUATIONS
98
whose
we
solution
To
avoid
constants.
discussions
in the
of
When
m exist,
let
us limit
notation,
case in which
the
to
$
does not consist of
Consider the system
w
Ai,---,Ap,
(8)
25.
26 by inspection.
of
tedious
now
ourselves
indicated in
is sufficiently
Mi
find the
(^y
-----\-t*P yP )
v -\
unknowns MU
(9)
-,
uq
yu
\
,
yp
\
w.
The totality of forms which vanish for which annul no separant is the system
all fi
solutions of (8)
of
28.
other closed essential irreducible system held by (8)
Every is
held
by some separant.
We
rearrange ihe unknowns
MI,
-,
and apply the process
uq
\
w\ yi,
-, yp
67 to resolve
of
order
(9) in the
,
(8) into irreducible
We
test these irreducible systems to see whether systems. they are held by the separant of some Ai, and pick out JS"5 which are held by no separant. those, say JSi, ,
,
As
has only one essential irreducible system which is held by no separant, there must be one Si which holds all other Si. To find such a Si, we need only find a Si whose (8)
For, let the basic set of Si, does not hold Sj, the separant of some form in the basic set of Si, must hold Sj, so that Sj
basic set holds
hold
S9
-
,
,
cannot hold
28
all
.
2l%
other Si.
If
2\
Thus,
if
^
does not hold every Si, no
Sj can hold every -,. Si will have a basic set
R
is an algebraically irreducible form in w and which the HI and in which JRi, 1, By ,/>, introduces yi.
in
/=
=
determines yi rationally in 28, 29 each equation J? is a resolvent for S. and the m and
R=
w
V.
CONSTRUCTIVE METHODS
99
A REMARK ON THE FUNDAMENTAL THEOREM results of 65-67 furnish a new proof of the 7 that fact ever} finite system of forms is equivalent to a finite number of irreducible infinite systems. Using the lemma of
70. The
7,
we
obtain the theorem of
13.
This
new
proof of the
fundamental theorem appears to us not to depend on Zermelo's But only that part which is stated above has been axiom. demonstrated on a genuinely constructive basis.
JACOBI-WEIERSTRASS CANONICAL FORM 71. Let set.
We
2
be a closed irreducible system with (1) for basic be the prime system for which (2) is a basic set. build a simple resolvent, 0, for ^, with
Let
A
R=
w
#1
the en being integers. (10)
ap zp
-
1
We
have
MiZi
N-t
0,
i=i,...
9
p
9
where the MI, Nt are simple forms in w and the v<. Let & be the totality of forms which vanish for the common solutions of 2 and
Let R go over into a form R' when the vt, Zi, are replaced by the corresponding u^, y^. Similarly, let (10) go over into (11)
M' y tr -N' i
=
i
Then R' and the first members of shown that 2 consists of all forms for all solutions of (11) and R' of R' and the Ml do not vanish.
=
0.
(11) are in in
.
It
can be
the u i9 yt which vanish for which the separant
Suppose that there are no m. In that case, the system (11), with w defined by R' 0, when converted into a system
=
of the first order,
by the method
of adjunction of
unknowns
used in differential equation theory, assumes a form equivalent to the Jacobi-Weierstrass canonical form.* *Forsythe, Treatise on Differential Equations,
vol.
U, pp. 11-14.
CHAPTER
VI
CONSTITUTION OF AN IRREDUCIBLE MANIFOLD
SEMINORMAL SOLUTIONS Let 2
72.
V'
be a non-trivial closed irreducible system in '
*
'
2/i
^
which
for
(1)
each
A
of class q-\-i, is a basic set.
.4 t
a
called solution
A
of (1) for which no separant vanishes will be normal solution of (1). By 66, every normal
solution
of
a solution of
(1) is
(2)
of
y 19 ..-,yp
HJ,---,^; for
(1)
seminormal area
-2.
solution
33 in
which if
some separant vanishes
there
a
exists
which the functions
set
of
will
points,
be called
dense in the
in (2) are analytic, such that,
given any point a of the set, any positive integer m, any *>0, there exists a normal solution of (1), u 19 -
and
-
,
yp
>
analytic at a, such that I
i
The
Mffcfo)
=
1,
HI* (a) ,
q\
|
j
=
,
!
1,
results of this section
and
yjk(a
yjk(a)
-,
k
p\
of
=
0,
,
m.
73 will show that the
existence of a single point a, as above, implies the existence That is, a solution for of a set of such points dense in 39.
which some separant vanishes, and for which a single point a exists, is a seminormal solution.
We
shall prove that if
morphic in
O
is
a form
until coefficients
mero-
21, the coefficients not belonging necessarily to ^, 100
CONSTITUTION OF MANIFOLDS
VI.
and
G
if
1Q1
vanishes for all normal solutions of
(1),
then
G
vanishes for all seminormal solutions of (1). More generally, we shall show that G vanishes for every solution (2) for which a single point a, as above, exists.
Multiplying G by a power of (x that the coefficients in G are substituted into
zero at a.
This
G
(?,
is
a), if
necessary,
analytic at a.
we assume
When
(2) is
becomes a function y(x) of x which is G vanishes for all normal solutions
because
and because of the w, e property of a. Similarly,
is
identically zero,
and
G
vanishes
for (2). If
we
G
forms
restrict ourselves to
with coefficients
we
in
see that every seminormal solution of (I) is a solution of are going to prove that the manifold of 2 73*
We
composed of the normal solutions of
(1)
e?,
2
.
is
and of the seminormal
solutions.
In particular, reducible
form A
of an algebraically irnormal solutions of A and
the general solution
the
is
composed of of the seminormal solutions. Let At be of order n in yi. Let 6V be the separant of At. For every yis with s>n, in a normal solution of (1), we
have an expression (4)
yis
= F
where B is a form of class at most q-{-i and of order at and where F is a product of most TJ in i/;, j -, i, 1, forms The of ,&. 5i, powers
=
Fy
(5)
B
is
belong to 2.
Let to (1)
be any integer greater than every r*. We adjoin 1 all forms (5), for i p, with * <; m. Without
m
=
,
,
going through the formality of replacing the u^ yij by new symbols, let us consider the forms in (1) and (5) as a system Q> of simple forms in the ?#, yy.
That
is
any set
of analytic
ALGEBRAIC DIFFERENTIAL EQUATIONS
102 functions of
of
.?
Uij ,
We
We
which annul the forms
?/;;,
of
will be a solution
do not ask, for instance, that yij+i be the derivative
&
shall prove that the totality
vanish for
solutions
all
Let a prime system. annul no Si. By (5),
of
GH vanish we
of simple
which no
for
Si
forms which vanishes,
for all solutions of
is
which
have, for these solutions,
>
where B and B2 involve no yy with j n and where JFi and jP2 are power products of the $. Then JBt J5 3 vanishes t
above solutions.
for the
By 65, (1), regarded as a set of simple forms, is the basic set of a prime system (even after new mj are introduced). Then (1)
B
either
l
vanishes for
which annul no
Then
O
We
Si or i? 3
vanishes for
separant so that
solutions of the simple forms does. Suppose that BI does.
all
which annul no
of
solutions
all
tt is
prime.
prove that, given any solution of 2, the t(/, appearing in #>, obtained from the solution, constitute
yij
shall
a solution of Q. of
Then
(1).
This
if
O
obvious for the normal solutions
is
is
a
a differential form in the
form
in
yj,
holds
Ui,
,,
considered
(?,
as
This proves our
2'.
statement.
Now
be
let (2)
a solution
of
2 which
annuls some St.
Consider the corresponding solution of Q. By 64, there is a region 2T such that, given any *>0, we can find a solution mk, yjk of ., with no Si zero at any point of 21',
such that
be
to
(3) holds at
taken
throughout
Now
if
that
so
every point a of the
coefficients
We
SI'.
in
(1)
suppose
are
81'
analytic
21'.
a
is
any point
of
21',
the
yjk (a)
tfc(a),
in (3)
furnish initial conditions for a normal solution of the basic set of differential
forms
(1)
(65).
there exists a normal solution (3)
with the solution
(2).
te,,
Thus yj of
any a in 21', which satisfies
for
(1)
CONSTITUTION OF MANIFOLDS
VI.
of
1Q3
We repeat the above procedure, using 2m and c/2 in place m and e. We find a region 21", in 21', every point a of
which can be used as above. find a region 2f
of regions 2I (0
"
There
.
a normal solution of
As
there
(2)
is
an a
is
4m
Employing
and
we
continue, determining a sequence least one point a common to
and any m, there
>0,
is
analytic at a, for which (3) holds.
(1),
every area in which (2)
in
f/4,
at
is
Given any
these regions.
all of
We
in 21".
a seminormal solution of
(1).
Our
result
is
analytic,
is
proved.
very likely that the set of points a consists of all at which the functions in (2) are analytic, with the points One might ask, also, possible exception of an isolated set. It is
whether every seminormal solution can be approximated uniformly in some area, with arbitrary closeness, by a normal solution. A negative answer would certainly be interesting.
These questions need more attention than
we have been
able to give them.
Consider the form in the unknown
Example.
?/,
algebraically irreducible in the field of all constants because, when equated to zero, it defines y s as a two-branched It
is
function of y and
Equating
A
y.
to zero,
we
dx
find,
y
for y 4 0, -
\
y
the solutions of which are given by
y
(6)
=
b e ll ~*>
and
with
b
above,
The
and is
c
constants.
included
solutions
The
among
these.
with
6^0
(6)
solution
are
all
y
=
0,
normal.
suppressed
From
the
b stays fixed in (6) at a value distinct from zero, while c approaches GO through positive values, y approaches
fact that
if
ALGEBRAIC DIFFERENTIAL EQUATIONS
104
& uniformly in any bounded domain, we see that the solutions b with b ^ are seminormaL Consider the solution y
=
Let & have any fixed value distinct from zero. By a sufficiently small negative number, we can make taking the second member of (6) and an arbitrarily large number
y
0.
c as
x
of its derivatives small at pleasure at
=
that y
is
A
solution of
is
=
0.
This shows
a seminormal solution and that the general the whole manifold of A.
Of course, by taking 6
small in
sufficiently
we can
(6),
approximate uniformly, with arbitrary closeness, to y
=
0,
normal solutions, in very arbitrary areas. But the discussion above shows what might conceivably happen
by means
of
in other examples.
74-
We
form not
can extend the preceding results. Let 2. It can be shown, precisely as in
F be
any
13, that
in
any solution of 2, there exists a set of points, dense such that, given any point a of the set, any positive integer m and any c >0, there exists a solution ti, ,%>, if
(2) is
in S3,
analytic at a, (3) holds.
for
It follows that
if
which
O
F
does not vanish and for which
a form with coefficients meromorphic
is
in 21, the coefficients not belonging necessarily to vanishes for every solution of 2 with
G
3^,
F^Q,
and
then
if
O
vanishes for every solution of 2.
ADJUNCTION OF NEW FUNCTIONS TO 75.
2
Let
of
be a non- trivial closed irreducible system.
^ not to consist purely of constants, we shall study the circumstances under which 2 can become reducible through
Assuming
the adjunction of
replacement
The
Pw
new
by a
functions to field S\ of
that
$r,
which
o? is
is, through the a proper subset.
functions of $i are assumed meromorphic in SI. form a resolvent for 2, relative to $, using a form
We Pw Q t,
of of
yi
1
as in
and
Q
28.
w which
for
which
Let
be the system of
vanish for
P^
0.
all
common
Listing the
all
forms in the
solutions of
unknowns
2
and
in the order
VI.
CONSTITUTION OF MANIFOLDS
Mi,
we
W\
Uq\
*,
?/i,
,
yp
1Q5
,
take a basic set
A, Ai,
(7)
&, with
for
,
Ap
A
S
algebraically irreducible in a resolvent for 2.
is
Then
r
.
4
=
Suppose now that the irreducible factors of A in 3^ are Then each J?i is of the same order in w as 4. jBs. J8i, ,
For, let r represent the order of A in w. If the coefficients of the powers of wr in A all had a common factor in e^,
common
they would have a
factor in
and
cF,
4
would be
reducible in S.
Consider the systems Bj, Ai,
(8)
/
=
1,
-,
Let
s.
.,
Ap
be the totality of forms in &i which
/
vanish for every solution of (8) which annuls no separant in (8). Then is irreducible in Si. Let 2j be the system / of those forms of to Si
2j
9
We
is
some
in Si
which are free of w.
Then, relative
2 holds every 2j that no 2h holds and that every solution of 2 is a solution ^ h, 2 2a will be a decomposition of 2 Thus, i9
shall prove that
any 2k with of
12/
closed and irreducible.
2j.
9
fe
,
into essential irreducible systems.
,
Since
-A _
8 -
-
T>
Jj2
.
.
.
I?
j$s
^i --
I
f-
..
every normal solution of A is a normal solution of some Bj. Thus every normal solution of (7) is a solution of some /. Hence a solution of 2 obtained by suppressing w in a normal solution of (7) is a solution of some 2j. Every is obtained by a suppression of w solution of 2 with P 4 in some solution of 12. Suppose now that some solution of 2 is not a solution of 1, ., s, which does any Sj. Let Cj be a form in 2j, j C8 does not vanish not vanish for the solution. Then Q But Ci C8 vanishes for every normal for the solution. -
-
ALGEBRAIC DIFFERENTIAL EQUATIONS
106 solution for
of
solutions
all
Thus every
2
of
by
JBj.
hence, reducible in S
and
in
also,
P!,
A
Hence
.
By
0.
74,
it
vanishes
it
holds 2.
a solution of some 2).
is
separant of A,
Then
P4
with
2
holds
it
72,
solution of
the
If 8,
divisible in
By
(7).
?,
were in some ty, it would be S would have a common factor and A would be algebraically
r .
Consider any form which annuls neither
Tof Any solution of (8), for some,/, S nor any separant in (8) is a normal .
%
so that T and annuls T. Hence ST is in Then every form of 2 is in -/, so that 2; holds 2/. Then the remainder 1\ Suppose that P is in some -0,. ( P with respect to (7) is in 2y. Then P, is divisible by fy,
solution of (7) in /Jy.
is
of
and
yi
possible,
P
r
have a common factor in since Pj is of lower degree than A
and I\
not in any
is
Now Pw
This
.
Q
in ?r r
is
im-
Then
.
Q. in
is
every
/.
It follows easily that
the totality of forms with coefficients in those common solutions of ~) and
r
t
Pw
fy is which vanish for
for which P^O. where ft^^ ^hen -r4 would hold /4. Then Bh would be in /4 and would be divisible by B k Then A would have a double factor in $1 and hence would be reducible in c?. Thus no 2^ can
This means that
if
held some
J2&
Q
2\.,
.
hold a
2
fc
with k 4-h.
Thus, /or
^
to be reducible relative to Si
of 2
sufficient that the resolvent
reducible in
We
see
^
of
=
$r
necessary
and
be algebraically
2
a resolvent for J/.
is
into
unknowns and the sum
n
Thus
in the
i, every systems in -, u q as arbitrary 31 is the same for \-rp of
irreducible
essential irreducible system will all
it is
.
that Bj
decomposition
,
relative to
-\
have w t
,
of the irreducible systems.
INDEGOMPOSABILITY AND IRREDUCIBILITY 76. in
Let
2 be an indecomposable system of simple forms We y n the domain of rationality being a field r
2/i
>
>
.
VI.
CONSTITUTION OF MANIFOLDS
prove that, if ~
shall
it
forms,
is
considered as a system of differential
irreducible in
is
107
*&.
Let A be Let (1) (with the unknowns relettered) be a basic set for A. Let G and holds 2. Let G\ and be differential forms such that with respect be respectively the remainders of G and
We
the
assume, as
we may,
that
of simple forms
totality
-2
is
non-trivial.
which hold 2.
GH
L
H
Then G\H, holds 2.
to (1).
We
H H
prove that one of
shall
6ri,
HI
is
identically zero.
Let GI and HI be arranged
Suppose that this is not so. as polynomials in the uij with
j>Qj
the coefficients being understand that no coefficient ?/!,--, yp is identically zero. The coefficients, being reduced with respect As 2 is indecomposable, there is to (1), cannot hold -S. a regular solution of (1) which annuls no coefficient. Let
simple forms
in
.
We
a be a value of x for which no coefficient, and no separant
For x
=
and for the values of above solution at a, GI and HI become i, -, yp in the polynomials y and h in the mj with j>l. Let numerical values be assigned to these mj so that neither// nor h vanishes. or
vanishes.
initial,
a,
-
We
now
construct functions Wi,---,?^, analytic at a, in the above solution and
whose values at a are the values whose derivatives appearing in G values assigned to the
u\j
regular solution in which values used above. Then
above. all ie#,
and
H
have,
For these ut, yi in GiHi have,
GiHi cannot vanish
at a, (1)
the
has a
at a, the for this regular
solution of (1).
Then G vanishes for all But every solution (2) of 2; can be approximated uniformly, in some area, by a regular solution of (1). The derivatives of the functions in (2) which appear Thus, let
G
l
vanish identically.
regular solutions of (1).
in
G
will be
approximated by the corresponding derivatives Thus G vanishes for (2) and holds ^.
in the regular solution.
Then 2
is
irreducible.
CHAPTER
VII
ANALOGUE OF THE HILBERT-NETTO THEOREM THEORETICAL DECOMPOSITION PROCESS ANALOGUE OF HILBERT-NETTO THEOREM In 1893, Hilbert, extending a result of Netto for 77. polynomials in two variables, proved the following remarkable be polynomials in y, theorem. Let -, a/-; 6, yn with numerical coefficients. Suppose that b vanishes for every a r all set of numerical values of y^, yn for which a. ,
,
-
-
,
,
Then some power of
a linear combination of the aij with polynomials in y\ y n for coefficients* The Hilbert-Netto theorem holds, with no modification of
vanish.
b
is
,
,
the proof, for simple forms. forms in y\ yn such that ,
,
If
G
Fif
,
Fr
\
G
holds the system
then some power of G is a linear combination of with simple forms for coefficients.
Assuming the following
THEOREM.
Let
foregoing
Fly
,
result,
F G r
;
we be
shall
are
F
,
establish
Frj Fr ,
the
forms in
F
r.
Then
and a
certain
'
>
,
J?\,
differential
y\* yni such that G holds the system JFi, some power ofG is a linear combination of the F{
simple
.
,
number of their derivatives, with forms for coefficients. 78. The above theorem will be easy to prove, with the help of an idea taken from Rabinowitsch's treatment of the algebraic problem, after we have settled a special case.
*A very simple proof is given by A. Rabinowitsch, Mathematische See also van der Waerden, Moderne Annalen, vol. 102 (1929), p. 518. Hilbert Algebra, vol. 2, p. 11 and Macaulay, Modular Systems, p. 48. gave results more general than the above. 108
HILBERT-NETTO THEOREM
VII.
F
Suppose that the system
show that unity
shall
if
Fr
,
109
We
has no solutions.
a linear combination of the Fi and
is
their derivatives, with forms for coefficients.
We assume that unity has no such expression and force a contradiction. First, we shall show that there exist n power series (1) i
coi
=
1,
+ en (x
a)
+c
2i
(x
a)*H
,
which, when substituted for the yt, render The series obtained may have zero radii of The derivatives of the series are thus under-
n,
,
each Fi zero.
convergence. stood to be obtained formally. In substituing the series for the yi into a form, we use the Laurent expansions of the After these formal solutions coefficients in the form at a. are
we
secured,
solutions
79.
We
and
able
are
yi
the
consider
Fr
be
shall
which the
in
show that there
to
analytic
actually
forms
of
system
of
exist
functions. of
consisting
derivatives -Fi , orders, writing the forms of the infinite system, in any order, in a sequence ,
of
their
HI
(2)
Similarly,
we
write
(3)
J?2
,
j
H
g
-
.
,
yy, arbitrarily ordered, in a sequence
all
*i
>
all
>
2
>
>
Zg
,
Each Hi will now be considered as a simple form in The domain of rationality will be r. To show the existence of the formal solutions (1), suffice
to
the
it
Zi.
will
a set of numerical values for the ^, and at which the coefficients in the Fi are analytic,
find
a value of x
which make every form in (2) zero. Consider any non-vacuous finite system tf> of simple forms Hi taken from (2). We shall consider the unknowns in
78.
of the
Hi
in
in contradiction of
HO
ALGEBRAIC DIFFERENTIAL EQUATIONS
We construct, in the Let q be any positive integer. 2 -, z q q of simple forms in z 1? following manner, a system 2 zq is to belong to in ^i, A simple form g if there holds.* exists a system tf>, as above, which Every 2q contains the form 0. When q is so large that an Hi exists .
K
,
K
j>q, 2Q
involving no
q
with
for instance,
it
will contain Hi.
We
shall prove
have other forms than 0;
will
2q which 2q
for every #,
that,
need consider only the case in
has solutions.
7, there exists a finite subsystem of
By
K
(4)
2q
which
With
holds.
O
a system
forms
of
t
19
(2)
..-,
We
has non-zero forms.
2q
,
Ks
each
Ki,
which Ki
there
associated
is
The
holds.
totality of
A
of forms (2).
forms in
,
,
Now A
,
2q
A, 2q
simply closed. Evidently, for every q, 2q is contained in 2? +i and consists of those forms in 2 q +i which are free of z q i.
holds
80.
holds
For each
g,
is
2q
let
be
decomposed
essential
into
prime systems .--, lit.
//!,
(5)
Then 2g consists of the forms common to all /T^.t Let H' be any prime system in the decomposition (5) of 2t We are going to show that there is a prime system 77 " in the decomposition of 22 whose forms free of z% constitute //'. .
Let AI
(6)
be the decomposition are
free
common
of
z2
(5) of
constitute
-
,
2t
.
A
v
Those forms
a prime system
^.
of
Ai which
The forms
to
Vi,---,^
(7) *
t
-
,
may
It is
involve
Zi
not in
K.
Of course
K
need not be in
(2).
unnecessary to express, notationally, the dependence of (5) on q
.
HILBERT-NETTO THEOREM
VII.
]11
the totality of forms in 22 which are free of ^ 2 that into prime systems. (7) is a decomposition of
is
,
is
^
Then
W
-\.
For,
^
were im Again, if some solution of not a solution of any ^, we could find a form Si in each Wi which does not vanish for the solution. *Sr Then Si ^i holds each
firstly,
,
which
in
is
^,
every
hence
in
21? would
not vanish for
the solution.
Thus the decomposition (5) of -\ is formed from (7) by suppressing certain ^-. Then some *Pi is identical with //'. This means that there is some prime system //" in the deSimilarly, there
28 whose
of
~ 2 whose forms
(5) of
composition
free
of z
constitute
//''.
a prime system 11'" in the decomposition
is
forms free of
z%
constitute //".
We
(5)
continue,
way, forming a sequence
in this
n',n",...,nw,....
(8)
We
81.
which form
is
now form
a system 12, putting into tt every form contained in any of the systems (8). Any particular
in 12 involves only a finite
number
of
unknowns.
We
are going to find a value a of x for which the coefficients in the Fi are analytic, and numerical values of
the
Zi,
for
which every form
in 12 with coefficients analytic
Since every Hi of (2) is in -C2, every Hi will vanish for the values found, and we will have the formal at a vanishes.
solutions (1).
There may be a that Zi
Zi
exist,
Zi
such that every form in 12 which involves some Zj with j ^ i. If such
effectively, also involves
we
select that
minimum, and
ut no non-zero form
call it
(not HI) such that
one of them whose subscript is a It may be that there is some Zi
.
of 12 involves only MI
and the new z%. If such Zi exist, we represent by u 2 that one of them whose subscript is a minimum. Continuing in this way, we form a set of unknowns m, which is either vacuous, finite or countably infinite, such that no non-zero form in & involves only the IH, while every zj which is not a m appears in a non-zero form involving only that Zj and the m.
ALGEBRAIC DIFFERENTIAL EQUATIONS
112
We
order arbitrarily the Zi which are not among the Ui, The sequence of vi, for all that we calling them vt v 2 etc. ,
,
can say offhand, may be in
assume,
what
modifications
only
trivial
their
number
or
finite
that the
follows,
of
countably
language are necessary when
finite.
is
In using the terms "initial", "remainder",
From among v\
minimum. only v$, to
AI
Vi
all
There exist non-zero forms in which involve and the m, and which are reduced with respect
From among
.
we
precedes every vj. non-zero forms in & which involve only select one, A lf whose degree in v t is a
we
ui,
below,
etc.,
m
shall understand that every
and the
We
infinite.
are infinite in number;
vi
degree in v2
is
we
such forms,
all
a minimum.
select one,
Continuing,
A% whose
we form an
infinite
sequence
A lf A 2
(9)
We
are going to
such a
way
Aq,
-,
,
show that
it
is
possible
to
form
Iq
of
Aq
that, for every q, the initial
only the Ui. This is true automatically for It been able to arrange so that TI,
in
(9)
involves
Suppose, then, that we have Iq -\ involve only the m.
.
,
B
be any non-zero form in -, v q involving vi, A g -i, and of and the ?i,-, reduced with respect to AI, as low a degree in v q as it can be with these conditions.
Let
,
,
The system vl9
,
v q -i
system. This the system of
of
m
because
is
all
in
W
The
we we
=
Oi
VH
A q -i
is
only a prime
is
=
a basic set for for
h
is
which involve only the
77<
A q -\
E
,
contained in some 77U> and
forms in that ,
the a
B, A if is
stated Ut, v. Then At, construct a simple resolvent (10)
which involve
forms in
all
and the
<2>,
We
with
flg-l Vg-l,
being integers. initial
Q
of
B
replace the vi in get a relation
is
not in
Q by
,
hence not in
their expressions in terms
When of
w,
H3
HILBERT-NETTO THEOREM
VII.
(ID where where
P
and 8 involve
w
is
w
und the m, the relation holding,
as in (10), for every solution of
Then
for every solution of
P
SQ
(12)
P
not divisible by
is
L
L
not zero, and free of w, that
is,
From
and (13) we see
(12)
w
and for
first
w
is
member
replaced
of (14)
V
in
B
Let
not in
identity
involving only the
^e^
=
in (14)
by
and
U
Then C is in Q and The remainder D of s in
D
for
vq
=
expression (10), the 0. We have thus
UQ + V,
s in v q
.
=
is
of degree s in v q ,
with respect to
(7,
Its
.
with
A i9
L ,
for initial. -d^-i,
will
will involve only the ui. This proves our statement re-
initial
in (9).
what
Let
C
A5
lative to (9) and, in
its
a form in the unknowns in
be of degree
be of degree We can use
0.
becomes a form in
L with
is
that, for every solution of
L
MSQ if
As Q
as in (10),
(14)
Then,
as in (10).
is
Thus, we have an
R.
$4
=
MP + NE =
(13)
with
w
if
with
follows,
we assume
that every
Iq
involves only the m. 82. are going to attribute constant values to the Ui in
We
such a
way
that each Ii becomes a function of
not vanish identically. Each /,- has at most a type Ui
h, h constant.
finite
Thus the
number
of
x which does
factors of the
set of polynomials Wi
h
ALGEBRAIC DIFFERENTIAL EQUATIONS
114
which are factors of one or more
Then
let
be a
ci
HI
we
If
constant such that no
put
^
the
in
(\
or countable.
/* is finite
I
z ,
J
divisible
is
z
each
by becomes
It
a polynomial J*, free of u and not identically zero. Similarly replace n g in the J/ by a c 2 so that no J} vanishes {
we
Continuing, we replace all m by constants in thai each // becomes a non-zero function of ,r.
identically.
such a
way
83. Let 33
the Fi are
be an area
analytic.
which the
in 21 in
Then every
H
=
q
in (2)
coefficients in
has coefficients
m
fixed as in 33. The equation A 0, with the or determines more functions of one vi, 82, x, analytic in some area Si in 33. Let one of these functions be selected, and substituted into A*. Then A 2 gives one or more t- 2 analytic*
l
in
=
analytic and the
in
contained in
33 2
selected above, into
/; x
using an area
33 3
in
33 2
.
We
33i.
and solve A$
^1 3
We
,
substitute such a
continue,
annul those forms in
for v s
?i,
uq
,
m in AI
to-
,
Aq
,
,
,
and
a vq
finding
a 33g for every q. For any g, the functions gether with the constant values attributed to the
t- 2 ,
which involve only the unknowns
,
in
Aq Let a be a point common to all areas 3v Then, a, the values of the Vi at a and the constants selected for the M, annul those forms in tt whose coefficients are analytic at a *TI
,
.
,
.
In particular, the Hi of (2), vanish for the above values. This proves the existence of the formal solutions (1) of
^i,"-,-Pr.
(15)
84.
We
now prove
shall
that (15) has analytic solutions.
It is not difficult to see that the results of
23, 24 hold
also those of
when
7-14,
a solution of
2
is
and
defined
as any set of series (1) which formally annul every form in
With into
know ?
,
(16)
this
closed
new
essential
of course ,
definition of solution, let (15) be
from
systems 2l9 78, that the Fi are not
irreducible
.
decomposed ,
28
.
all zero.
u q be a set of arbitrary unknowns for 2i and
<4i,---,4,
~
We Let let
HILBERT-NETTO THEOREM
VII.
115
each At introducing ?//, be a basic set for 2\. the separant, and It the initial, of At.
We
are
show that
to
going*
has
(16)
Let
analytic
S\ be
regular
solutions.
in
Suppose that when the At are regarded as simple forms the yy, H J which they involve, they have a solution, }
consisting of analytic functions, which annuls no Si or /f. Then the values of the u,j, yij at some suitable point will furnish
initial
an
for
conditions
regular solution
analytic
of (16).
Now,
if
T
vanished for of
T
But
Sp
TI
y,j
of (16) considered as a set
Jp
we would have,
theorem as applied (17)
Si
solutions uy,
all
forms,
simple
=--
h
using the Hilbert-Netto
to simple forms,
-
CVli-f..-
would continue
an identity
+CP A P
.
to hold for all formal
power
series
solutions of (16) considered as a set of simple forms.
Then
(17)
the basic set of differential forms (16) would have no regular
power
series solutions.
This shows that (15) has analytic solutions. We have reached a contradiction which proves that unity is a linear combination of the Fi, and of a certain number of their derivatives, with forms for coefficients.
85.
We now
complete the proof of the theorem stated in
adjoin an
unknown
77.*
We
z to the
y^ and consider the system
of forms
zG-\, Fl9
,
F
r
which evidently has no solutions. Let there exists an identity in the Zj, yg,
m
'-<>
< 18 >
where the '
Cf.
Cj, D,j are
Rabinowitsch,
j
;
forms in
loc. cit.
in
,
K=
r
z, y\,
zG i
-t
,
yn
.
\.
Then
ALGEBRAIC DIFFERENTIAL EQUATIONS
116
If we replace z by derivative of I/O, the
1/0
in
first
sum
and each
(18)
by the ;th
gj
in (18) vanishes.
We
find
thus an identity
where the Ey are forms Then
and
this establishes
Example.
We
86.
J5\,
our theorem.
consider two forms in the
=
1.
J^ has no solutions.
We
y\ F,
differential equations
unknown
y,
have
84 that if a system of algebraic y n has formal power series
follows from
It
=
yi
F,
The system
in
in y\,
,
solutions j then the system also has analytic solutions.
One might ask whether a system 2 which is irreducible when the solutions y^, y n are understood to be analytic functions, remains irreducible when the yi are allowed to be formal power series. The answer is affirmative. Let Then one of G, hold 2 according to the second definition. ,
OH
H holds
2
for the
G
does.
with the same manifold
(first
Suppose that
definition.
first
Let (19)
be a
JSi,
finite
subsystem of as 2.
definition)
2
,
B
a
Then some power
of
G
bination of the Bt and their derivatives.
is
a linear com-
Then G holds 2
for the second definition.
THEORETICAL PROCESS FOR DECOMPOSING A FINITE SYSTEM OF FORMS INTO IRREDUCIBLE SYSTEMS 87. in
yl9
We ,
deal with any finite system
yn .
2
of differential forms
Let p be any positive integer.
We
denote
HILBERT-NETTO THEOREM
VII.
2W
the system obtained by adjoining derivatives of each of its forms.
by
p
U7 to
2
the
first
the forms in 21 & are regarded as simple forms in which they involve, 21& goes over into a system
When the
yij
AW
For domain of rationality, we use of. the method of Using 55-60, we decompose A &, by a finite number of operations, into essential indecomposable of simple forms.
(
systems !,... ,0> r
(20)
.
Let the forms in the
Let any Wi which is held by some Wj with j 4 i be suppressed. This can be accomplished by a finite number of operations ( 68). There remain systems
differential
forms
.
^i,---,
(21)
We of
2
say that, for
p
"V
sufficiently great, (21) is a decomposition
into essential irreducible systems.
88. Let
2i,---,2
(22)
be
a
decomposition
systems.
forms in
of
2
into
finite
essential
irreducible
When
the forms in the 2t are regarded as simple their y#, (22) goes over into a system of simple
forms
-,/.
/\,
(23)
Let us make any selection of t forms, one from each 2\-, and take their product. Let the products, for all possible selections, be Aij A% Ag 9
Then each At holds 2.
By
.
,
77,
if
p
is large,
some power
each At will be a linear combination of forms in 2W, with forms for coefficients. If then each Ai is considered as a simple form in its y#, of
and if
p
if
is
it is
represented then by
sufficiently large.
2?<,
each Bi will hold ^/ (p)
Let p be large enough for
this.
ALGEBRAIC DIFFERENTIAL EQUATIONS
118
We
held by some /} Let Cj be a form
shall prove that ^ach <&i of (20) is Suppose that #>i is not so held.
of (23).*
^
of /}, j
that
is,
,
,
does not hold
J3,-,
Then C\ ! CV, Then that JB cannot
which does not hold
t
.
-
{
.
t
This proves our statement. is held by some 2/. is held by some tyj. the other hand, each
hold s/ (p}
.
follows that each Wi
It
^
On be
1
some
false.
Let Dj be a form
momentarily,
Then G
-
*/{/,
J
=
1
-
,
,
Let
this
(we restore,
r,
suppressed '/{/) which does not hold ^\ does not hold Hence (r does not
the
A
in
.
^
7>V
.
Then, if G is considered as a simple form in its yij, does not hold ^W. This contradicts the fact that ^ (p)
hold 2. it
equivalent to (20).
is
Thus, for p sufficiently great, (21)
a decomposition of
is
2
into essential irreducible systems.
For the above process to become a genuine method of decomposition, it would be necessary to have a method for determining permissible integers p. This question requires further investigation.
In
89,
we
treat a special tfase.
Example st
{
&
is
1.
2
Let
be y\ 4//, in the unknown y. system
Then
equivalent to the y?
*y,
2),
2/iO/2
+2
2/i 2/4
?/2 2/3
2/12/3
+ + G>
+
2),
2/2(2/2
2),
2/3 (2/2
-
i) 2/p-i
-,
2/3
-f
2/p
(y
~2
)-
decomposes into the two indecomposable systems (24)
2/,2A
in the
_ unknowns
(24), that
(26)
(27) *
2/
?
2/u
2/a-
If
we
The unknowns are
y, yi, y*
y>
all
2/i>
ft
which appear
_
adjoin y 2/3+2/2
system decomposes into
2,
in (20)
and
(23).
(2/2
2) to
which are systems to (25) gives (27)
Thus
in y,
ylf y2
we
Continuing,
2,
y,
ya
.
and (28) give the decomposition
(27)
(26),
The same adjunction
y$.
,
and
y\-ly,
(28)
H9
HILBERT-NETTO THEOREM
VII.
the decomposition
find
A^
of
of
to
A^\ for
be,
P>2, (29)
y,
?/i,
(30)
y,
ylj
s/2
(31)
y?
If
we regard
,
y8
2,
4y,
yp
,
2/2
,
,
2,
yP
,
2/3,
,
Jtefi.
,
the last three systems as systems of differential
forms, (31) gives the general solution of y\ 4y, while (29) second is irreducible which a the solution 0, gives y
=
The system
manifold.
(30) of differential forms has
no so-
lution.
We
2 notice that the system of simple forms y\ 4?/, y% holds the system (30) of simple forms. This is in harmony with the fact that every O* in (20) is held be some 7) in (23).
Example
2.
Let
2
be the form y\
fact that its manifold is to be
y
=
an irreducible system.
\l(x If
VP A!
Then (32)
+ (p
^/ (1)
1)
4y*,
of
8
which, from the 0, is seen
,
and y
=
let
AI by
A r +i,
then
y^A v
yp-iA 9 H-----h (p
decomposes into y, yi 8
(33)
a)
we
and represent the rth derivative will be y\
4y 2
2/?-42/
,
^.
1)
2/2
Ap-i
+ Ap 1/!
.
ALGEBRAIC DIFFERENTIAL EQUATIONS
120
We now
examine
The adjunction
^/ (2) .
of
y^Ai+y^A^
to
+2
A$
(32) gives the single system
(34)
yl9
y,
The same adjunction
ys.
to (33) gives
0f-4y, A l9
(35)
and also
A*,
(34).
Let us examine
^
(8) .
The adjunction
of y* AI
1/ 2
A 2 -f
?/i
to (34) gives the single system
(36)
y, ?A,
in the
unknowns
y,
,
*/4
.
y2
The same adjunction
to (35)
gives
y\-y\
(37)
A,, A,, A,,
as well as the system, held by (36), obtained
by adjoining
y, to (36).
Continuing,
it
&
is
sition of sl (
not
is
(38)
y,
where q
is
prove that the decompo-
difficult to
-
yl9
yq
,
the greatest integer in
(39)
+ p/2
,
and
yJ
The system
(39)
of differential
of 2, while (38) (differential forms), is
1
forms gives the manifold whose manifold is y 0,
=
held by (39).
FORMS 89. Let
A
order in y,
how
ONE UNKNOWN, OF FIRST ORDER
be a form in the single unknown y, of the
and irreducible algebraically.
to determine, in a finite
of forms
Let (40)
IN
^1
whose manifold be of degree
m
number
We
shall
of steps, a finite
first
show
system
the general solution of A. in y l consider the system is
.
We
^,4 ,...,^ m -i 1
>
HILBERT-NETTO THEOREM
VII.
121
there Aj is the /th derivative of A. Let (40) be considered as a system of simple forms, and let it be resolved into finite t
essential indecomposable
There
systems.
be precisely
will
one indecomposable system, ^/, which is not held by S, the separant of A ( 73). Let the forms of A be considered now
Let
as differential forms in y. forms thus obtained.
We
shall
solution
know
2
manifold of
that the general solution of
What we have
in the manifold of 2.
2
solution of
We
the
be the system of differential the general
is
of A.
We
90.
that
prove
2
is in
observe that
to
A
show
contained
is
that every
is
the general solution of A. A holds 2. The solutions
A
of
not in
the general solution are solutions of 8. The common solutions of A and S are solutions of the resultant of A and S with respect to yl9 which
is
a non-zero form jR, of order zero in y.
R
which is then to show that every solution u of a solution of 2 is contained in the general solution of A.
It suffices
Let yi
Then
uj be the jth derivative of u.
= HI-
There exists an open region
2li
A
for
y
= M,
and an
A>0
such
that, for
x
(41)
and
in Six
0<|y
u|
every solution of the algebraic relation sidered as a function of (42)
yi
-
ti!
=
y and x,
= 0,
for y^ con-
given by a series
+ ap (y
tt)"+
ao(y
is
A
where the at are functions of x analytic in 2ti and where q and s are integers, s being positive. The particular series used
in the
second
solution yi used.
We suppose that, The system
depends on the particular we have s <J m. OQ does not vanish for every x.
of (42)
But, for each such series, in
each
series,
of functions
(43) is
member
a solution of A.
u,
uu
By
which we can approximate
-
-,
um
64, there is a region 21 2 in 2d in arbitrarily
closely
to (43) 9*
by a
122
ALGEBRAIC DIFFERENTIAL EQUATIONS
solution of
A
R
with
be
to
21 2
suppose
21 2
analytic throughout It follows that, if
A
equation
-
?*,
We
.
A
are
.
is
any point
in 21 2
has solutions analytic at
which
for
distinct from zero throughout 21 2 taken so that the coefficients in
the
,
differential
R^
with
J,
at ?,
from y, y m differ arbitrarily slightly at u m respectively.* such solution satisfies (42), in the neighborhood of J
-,
Any
,
,
Hence an appropriate choice of the series in (42).t there must be one of the series for which (42) is satisfied
for
by a solution of A with JK 4 as one pleases at 5 to u, such a
deal with a,,
at
4
We
see
not be
series
.
,
.
>0
that #
first
small
we
(42)
and with y, -, ym as close In what follows, we um to taken so that and assume -
at
?
if
y
Otherwise
in (42).
n
small at
is
MI
?/i
would
Differentiating
.
find
(44)
Replacing y*
H t in (44) by
?/,
- MI 1
are analytic in
bi
>0.
when y
u
Similarly,
2li.
=
S
The
\
zero at ?.
_
,
notice that, if m 2, could not be small at :
MWI
\
lmq__ m + 2
\
I
\
1
][
8
8
)m ,
(
s-
m+l
term of the second member
Hence the first exponent That is,
R
t
at $,
y
if
-
>
We
in
S *
at I for a solution of A, then
u
+ is
.
.
.
.
not
the series in (45)
positive.
Note that
H R
- tt)^"-"- +
find
coefficient in the first
must be
find
small.
is
we
q_/2 ?
&i (2/
u2
Otherwise y a
we
1
1
ym (45)
expression in (42),
- w) "- + 8
a\ (y
,
&
where the
2q/s
q
its
for a neighborhood of
.
at
HILBERT-NETTO THEOREM
VII.
123
.._+ 1>0>
S
so that
mq-
1 =m +i-i-^ ^
8
.9
Thus,
8,1 mm
^ q>s *
s<
and, as are
We
w, we have
now
able to
-,
\-
#>s
1, so that
In (42), the into differential equation over goes solution of
(46)
s
.4.
=
f-doc
q>s.
show that u belongs we replace y u by
a Q t*-*H H
h ap
the general
to
vs
Then
.
V+P- S + I H
(42)
.
member of (46) is analytic in v and x and x close to ?, then, if we fix z; as a small distinct from 0, (46) will have a solution quantity at not identically zero, and with any desired analytic at Since the second
for v small
,
,
number Then y u finite
=
of
Vs
derivatives as small as one pleases at will be small at while not zero at ,
,
.* ,
together with as great a finite number of its derivatives as one may choose to consider. Solutions of A, close to w, but distinct
from w, at
5,
cannot make
R=
0.
Thus, if u is a solution of S, as well as of jff, u is a seminormal solution of A and belongs to the general solution If u is not a solution of S. y r^-tainly belongs to of A. the general solution. *
Equation (46)
is
satisfied
the constant of integration.
by v
=
,
and
its
solution
is
analytic in
CHAPTER
VIII
ANALOGUE FOR FORM QUOTIENTS OF
LtrROTH'S
THEOREM
91. It is an important theorem of Liiroth that if a and ft are rational functions of x, then a and ft are rational functions of a third rational function, y, which, in turn, is a rational combination of and ft*
We
are going to prove the following analogue of Liiroth's
theorem.
Let a and
THEOREM.
and
(a)
ft
Y
is
of
The of
its
38) in
two
form
quotients
there
exists
a form quotient
be
are rational combinations of y
number of (b)
ft
Then
a single unknoivn y. in y, such that
(
y,
and of a
certain
a certain
number
derivatives,
a rational combination of
et, ft
and
their derivatives.
coefficients in the rational combinations are functions
x in o?. As to the degree
of uniqueness of y, we prove that if Y\ and Y* are two possibilities for y, then y* (#yi &)/(cj'i4~^) with a, by c, d functions of x in 3\ 92. We prove the following lemma. LEMMA. Let PI, -, Pm Q, R be forms in y, R not
=
-
+
-
,
identically zero.
Suppose that the relations ~ Pi(y)
~
Pi(z)
where y and z are analytic functions for neither of which vanishes, imply the relation
R
*
Van
Appel et Goursat, Fonctions Algtbriqucs, 2nd der Waerden, Modem Algebra, vol. 1, p. 126. 124
edition, vol. 1, p. 283.
VIII.
Then
THEOREM
LfiROTH'S
125
form quotient Q/H is a rational combination of Pi/Rj and of a certain number of their derivatives, mth
the
the
coefficients in
\
The values
and 2
1
of
m
will suffice in our applications
lemma.
of this
Let v^
vm
.,
\
w
be new unknowns.
vi-Pi(i=
(3)
Consider the forms
Rw
l,.-.,m);
Q.
32, the system 2 of all forms in the Vi, w 9 y which vanish for all solutions of (3) with-K^O, is irreducible. It
As
ig
easy to prove that any one unknown in
is
arbitrary unknowns. take vi as arbitrary unknown, for 2
We
A
(4)
-..,
2
is
a set of
and form a basic set
A m B, C ,
which introduces,
in succession, v2 *,vm w, y. are going to prove that is of order zero in w, and, indeed, that it is linear in w. ,
Suppose that
B
is
exist
because R(y)
w. Consider Such solutions
of order greater than zero in
regular solution of (4) with
any
,
B
We
is
not in 2.
R(y) ^ Without disturbing the 0.
Vi
the solution, we can alter the initial conditions for w slightly at some point and get a second regular solution with R(y) ^ 0. This would be contrary to the hypothesis of the lemma.
in
We
shall prove
now
that
B
is
linear in w.
Suppose that
this is not so.
Let AS, v2
,
In
,
-,
vm
(4),
.
we
A m be of the respective C be of order r in y.
Let
replace the symbols *
by
w
>
orders r t ,
,
r m in
ALGEBRAIC DIFFERENTIAL EQUATIONS
126
The remaining v# and
respectively. order,
by symbols
Then
(4)
Ui (see
we
\ji
in
replace,
any
65).
goes over into a basic set
E
F*,...,Fm D,
(5)
,
of a prime system.
Let
CV--,^
(6)
be analytic functions of the
m
when
zm
at
substituted for z t
2
least
,
,
we can
s m +i,
in
two
get
"^ and +1) dr Ci" which, with
E
we
have two
will
'
b
which annul
',
when
(5)
>
After
v(w+2)
v(m-f-l)
'
,
Fm
functions, 46).
(
,
i
v(m)
.//
D
,
of degree
sets of functions
,,, ^''
2
is
distinct
annul
(6),
D
As
45).
(
1
treating
F
and x which annul
fe2
,
,
fe2
substituted for the
No separant
Zi.
or initial in (5) is anulled by either set (7). be the remainder of with respect to (4). Let Let which are new Ui be taken, to correspond to the vu in
R
T
T
not in
Let
T
go over into a simple form ?7in the %, Zi. reduced with respect to (5), it will not be annulled
(4).
As
U
by
either set (7).
is
We
numerical values to
attribute
following way. cients in (5)
We
and
require, secondly,
that
U
in
x and
m
the
in
the
require the functions (7) and the coeffito be analytic for these values.
U
that
^^
$"+
and the separants
in
(5)
We
4-
0.
Finally,
we
ask
do not vanish for either
set (7), for these values.
The values chosen
furnish initial conditions for
which annul neither same vi in both solutions. of (4)
for both
same
in
solutions.
On
both solutions.
the lemma.
R
nor any separant.
Then
t' 2
,---,t' m
two solutions
We
are
use the
the same
other hand, w will not be the This contradicts the hypothesis of
the
LUROTH'S THEOREM
VIII.
B
127
B
PJR
in w. If we replace n in by the resulting expression in // must vanish This completes the proof of the lemma. identically. Let 93A(y) and B(y) be two non-zero forms in //, relatively prime as polynomials in the iji and not both free
Then
and
of
is
linear
w by Q/ R,
Let r be the maximum
y.
of
their
orders
in
//.
We
shall prove that
A(y)B(g)-B(y)A(g)
(8)
is
not divisible by any form, not a function of x, which does
not effectively involve yr
Suppose that there
=
A(y)
is
.
such a factor, C, free of y r
=
B(y)
J/^+.-.+.Vfc,
where it is possible that either 3/ or must be a factor, for every i, of
We that of Xj
shall
C
show that
involves only
we can
values to y,
C cannot
#<,*+
N
"+ A
is
*>
Then
zero.
be a form in z alone.
Since A(y)IB(y)
z.
is
Let
.
C
Suppose
not a function
assign two distinct sets of rational numerical yr in such a way that ,
AI
B
BI
2
where the subscripts correspond
A2
,
to the substitutions, is not zero.
As, by (8)
At
B (z)
B,
A (z}
B
A*B (z)
,
2
A
(z)
are divisible by C, then B(z) and A(z) must both be divisible by C. Thus C would have to be a function of x. Let C be of order s ;> in y. Let A(z)/B(z) and let
Wi
For any B(z)
^ 0,
=
Mi
rational numerical
and for which the
aNi,
values of
i
g,
-,
= =
0,
gr
for
coefficient of the highest
,
h.
which
power
ALGEBRAIC DIFFERENTIAL EQUATIONS
128 of
a
C
in
y8
does not vanish, the expressions wi will which will be a polynomial in the
common
factor,
S
coefficients in
all
yi,
have with
r .
Let \~VhWhj
where the will
are
Vi
iii,
rational Ui and
indeterminates.
and for rational
vt,
Zi
for
Then,
as above,
arbitrary
wu und wv
both be divisible by a polynomial effectively involving y8 resultant Q of w u and ivv with respect to ys must .
Then the
vanish identically in the m, Vi and x, y, yr -i with the if is obtained for the indicated substitutions omitted, by ,
Now on the
a polynomial in Since a depends effectively can find an infinite system of sets of numerical
is
Q
i/5
Zi.
.
we
Zi,
values for the ^, as described above, each set giving a distinct
Thus
Q is identically zero, even in have a common factor which is a polywith coefficients in S nomial in the M,-, tv, ?A and Thus, the expressions wt, with a indeterminate, must all have
result for
.
wu
Then
.
wv
and
r
.
,
a
common
a with If
factor d, which
8 were free
divisible
Thus d
by is
6.
a polynomial in y,
a,
A
with Hi, E,
F
yr-\
y
every Jlft and every Ni would be and B would not be relatively prime.
of the first degree in
Then, for every
,
.
of
Then
is
r
coefficients in
i,
.
we have
forms in
t/.
Then
(8)
has a factor
which is free of z. This is impossible, for the same reason which C, above, could not be a form in z alone. The
for
proof
94. 91.
is
completed.
We
proceed with the proof of the theorem stated in 1. When a and ft are both free of y, we take Y
=
In what follows, of y.
Let a
We
=
may P/R,
we assume write ft
=
and
Q/R.
that ft
and
with a
ft
are not both free
common denominator.
Consider the forms in y and z
VIII.
LUROTH'S THEOREM
129
P(y)R(z)-P(z)R(y),
which are not both identically zero. 2S be a decomposition Let 2ly
of
,
(9)
closed
into
From each 2if we select systems. a non-zero Oi which is of as low a rank as possible in z. assume, as we may, that each Oi is algebraically irreirreducible
essential
We
There must be some Oi which involves both y and z effectively. Otherwise, (9) would imply a relation of the ducible.
type
=
C(y)D(e)
For
=
0.
=
would become C(y) D(y) But (9) is 0. with y y arbitrary. Let GI, -, G9 Op involve both y and z, while G p +i, For i
y, this
=
satisfied for z -
,
of 2i is the general solution of Oi.
Let Gp+i, only y. Let
-
Then
M
solution
That
is
,
if
then, given
involve only
not in any 2i,
,
8
involve
i
<
because, in the general
p,
<J p, z can be taken almost arbitrarily. take the unknowns in Oi in the order 2, y,
i
we
any regular solution
any number
Gq +i,
and
z,
M= Gp+i'-.Gg.
of Oi,
is,
Oq
-
of its derivatives,
of
at
we can modify
(?,-,
some
point,
z
slightly,
and but
otherwise arbitrarily, and get a second regular solution of Oi. Op in z be r. Let the greatest of the orders of Gif , Let Oi be of order r in z f or i <[ m and of order less than r ,
for
m
p
.
Let
H=Gt...Gm
(10)
We
write
(11)
J5T
as a polynomial in zr
H=
.
.
Let
JV*-
There must be some ratio FJF which is not independent Otherwise some factor of F would be a factor of
of y.
ALGEBRAIC DIFFERENTIAL EQUATIONS
130
every Fi and // would have an irreducible factor not inThis would contradict (10). Let Ft/F be not volving zr .
independent of y.
K=
G m are .algebraically Hjdzr- Because GI, and none of them divisible by any other, and with respect to zr is not identicthe resultant U of Let
9
,
irreducible forms,
H
K
ally zero.
H
have the order y, z Understanding that the unknowns in with respect to H. We say B be the remainder of for are relatively prime polynomials. that B and If,
M
let
y
H
instance,
hold 2i.
than GI
B
were
F
Now in z.
divisible
not in
is
.
.
m+
.
d 2r
is
.
not in
2lf
K Fh M
some
since
of
is
it
would lower rank
#
+ 4,9
-
Gi
0m-i.
r
Each term after the first The first term is
in 2i
(?i,
2l9
Again,
K = -^ 0,
(12)
by
and
Thus the resultant
in
second member of (12) As is not in ~\
the
not.
Thus
K
.
is
M
are relatively prime.
V
of
H
and
jB
with respect to
zr
is
not identically zero. Consider the form
S
We
= UVFG m +i
0,.
can assign rational numerical values to 0, 2>_i S becomes a non-zero form T in ?/, and so that ,
that
so
_F/F becomes a form quotient y
in
y which
is
not a function
of x.
We
say that y as thus determined
satisfies the conditions
91.
of
95-
L(y)
(13)
Let
Let
?/
=
TR(y)Gq+l
...
G9
.
and y be two functions of x. analytic in some part which L vanishes, and for which
of SI, for neither of
R(y)
LUROTH'S THEOREM
VIII.
131
exist. For instance, we can use any // such and then take y Let a be any value y. L(y) of x for which ?/, // and the coefficients in R and every G i s are analytic, and for which L(yf) L(y)]-0. 1,
Functions
~y",
that
y
=
$-
=
t ,
,
H=
0, where equation 2V-! are replaced by the rational values used above, 5-, x by a and ?/ and its derivatives by the corresponding values for y at a, will determine /* distinct values of z r .* To each
the
By
definition
the
of L(y),
,
~
such zr will correspond a regular solution,t with y y, of some 6rj, ? <| m, which is not a solution of any Gi with ?
>m. A
(See (12)).
similar result holds for y. for
Now,
solutions
in
y equal to y or Let z, z.
values used above.
m< <
<
and A solutions
of these
2h
zr
,
\
With these
solution (y, z) or (y, 0) with ? j9 or q (y, z)
to y,
of ?'
will
have the same
the rational have, conditions for z, no at a,
initial
will
(9)
(9)
be a solution
There
6?.
will
be
of a
JSi
solutions
7*
which annul the form //. None M(z) and they will be the ^r _i y or y and with z,
(y, z),
solutions will annul
=
only solutions of (9) with y as indicated at a, for which Jf(#) 4 0. Thus, in the h soA annul the functions z must be the lutions (y 2-) which 7/, ,
,
same
as in the (y, z) which annul //. the numerical ?/, Then, for y y or for y
equation
must have the same h roots for zr
We may
=
7f let
=
for z r
^
the value a, of
H
x range over an 7
area.
.
Thus, in the ex-
zr -\ in (11), every ratio Fi/F, with z, pression for rational as above, must be, for every x, the same for y as for y. ,
Consider then /.
For
R(u)L(~y) and
for ft(y)
fl(y), similarly,
r(y) *
Consider that
fFor
S
is divisible
the order y, z.
=
by
F U.
we have
ALGEBRAIC DIFFERENTIAL EQUATIONS
132
92, Y is a rational combination of
By
number 96.
We
now
prove
and a certain
ft
,
with coefficients in
of their derivatives
a and
that
\
are rational in Y and
ft
derivatives.
its
In Ft and F, with
z,
zr -\ rational as above, let
,
y be
re-
placed by z. There will result two forms, At(z) and A(z). We wish to show that neither At nor A has a higher rank in z than H. is
it
Evidently, tained from
H by
higher rank than
For y and
H
if
/
is
the form ob-
then
z,
I
is
not of
in z.
z
let
,
,
23
Op taken y and
in
symmetrical
^
interchanging y and
Ei be the form which results from Oi when are interchanged. Then EI, -, Ep must be mul-
i^p,
tiples of 0^ is
show that
to
enough
in
some
order.
the interchange
z,
For,
since
of
arid z
y
(9)
in
an interchange in pairs of those No i can be converted into a 2j with with 2i systems. j>p. Otherwise 2* would contain a form in y alone or in ,
,
will accomplish
z alone.
This
is
for as
impossible,
was seen above,
either
can be taken almost arbitrarily in the general solution of Oi with i
y and
of
z
among themselves. If ^ Then JS* are interchanged, then Ei must hold 2). Jfy is not of lower order than Oj either in z or in y. By symmetry, J?i and Oj have the same order in z. Then J2i must Then 2\
-
,
,
Sp
are permuted
and
be divisible by Oj, so that, as Ei and Oj are algebraically irreducible, their ratio is a function of
x
in
r.
(?t has a greater rank in z than Oj \ii<m and j follows that I is not of higher rank in z than H.
As it
Let
A
t
(7
be a highest
= CW
t
We
and
A
are going to
which
We
is
common
= CW
> m,
factor for At and 4.
with F< and
F
relatively
Let
prime.
show that
not zero, equals first that J
H
observe
Taking the unknowns in be the remainder of L(z\
is
multiplied
by a function of x.
not of higher rank than Hin in the order y, z, we let
H
(see (13)),
with respect to
z.
Y
H.
LUROTH'S THEOREM
VIII.
Y
Then
and
H and
H are
Let
relative prime.
F
133
Z
be the resultant
with respect to zr Then Z is not zero. Let real or complex numerical values be assigned to does not vanish iden,-, zr-i in such a way that
of
.
UZF
y and
tically in
Let
derivatives.
its
=
Zi(y)
L(y)UZF,
where the substitutions just indicated have been made in U, Zj F. We observe that the coefficients in LI may not be in S
r .
Then, and if a
if
y
an analytic function for which Li(y)^Q,
is
a suitably chosen value of x, for which LI (y) ^ 0, the differential equation r -i as just 0, with z, -, at will determine h distinct functions z for which taken, a, is
H=
and for each
L(z)^Q
of
-
(*)
which one has (*)
(
-
/%)
Hence, for the chosen y and for each such
0,
one has ,7"= 0.
we modify
the numerical values attributed to z, 2>-i slightly, but arbitrarily, and make arbitrarily slight variations at a, leaving the higher derivatives in the values of y, , yr If
of
y alone, we
new
all,
of x, y,
mits
is
will still
have L^
(y)
functions z, which, with the
All in
as
,
,
we yr
and we
^
new
y, will
h
zero.
see that in some open region in the space ?r i, the equation J for zr adz, -
\
will get
make J
,
of the roots of the
equation H =
=
for zr Then, no repeated factors, J is divisible by H. As J in z and as J has no factors not of higher rank than all
H has
.
H
r
(not functions in S ) which do not involve zr fiHy with fi a function of x in S
J=
(
93),
we have
r
.
N
be the remainder for R(z) with respect to Jff for ot The resultant and JV with respect the order y, z.
Let
H
X
to zr is not identically zero.
for z,
,
zr -i in
D
Let becomes for these
not vanish.
X
and
in
Let rational values be substituted
FU
of
94 so that
XFU does XFU
be the non-zero form in y which R(y) substitutions.
ALGEBRAIC DIFFERENTIAL EQUATIONS
134
Now,
let
y and y be functions
for
which
W(y) D(y) W(y) D(y) +
(14)
0,
and which, when substituted for y and z in J, render Jzero. Let a be a value of x for which (14) does not vanish. The for z with y differential equation y, has h solutions z with ,--, 2V- 1 assuming the above rational values at a, and with jK(?) ^ 0. The A pairs of functions (//, z) thus
H=
obtained are solutions of
because
#= J/p,
(9).
H
which are solutions of
Then its
91
is
and
get
It
(9).
are
(?/,
z).
pairs (y, z) easy to see,
li
is
that the functions z in the
derivatives.
li
pairs (y, z}
It follows that
rational combinations of
As
Wt(y)IW(y) Wt(y)/W(y), the theorem of
y
proved.
The above everything
number
we
Similarly,
and of
are the same as those in the
and
=
y
proof,
essential
and the methods the
for
Chapter V, contain
of
construction
of y
in
a finite
of steps.
97. Suppose that we have two form quotients like y above, see immediately that ft is rational in terms Yi and ft. of y l and its derivatives and that y is similarly expressible
We
in ft.
We now apply We see that the
the method of
2
92.
Let v
=y
l9
=
v2
ft.
v* which vanish system Let A is irreducible. in for v i\ ft y 2 identically y Acin be a non-zero form of 2, of a minimum rank t?2 92 (YI can be made to correspond to P^/R cording to and ft to Q/R)j A is of zero order in v2 and is linear in t;2 Similarly if B is a non-zero form of 2 of a minimum rank in vi then B is linear in vi If we take A and B algebraically will be divisible by the other. as we each irreducible, may, This proves that Hence A is linear both in v and in va of each other. and linear are fractional combinations Yi ft
=
of all ,
forms in v^
=
.
,
,
.
.
.
CHAPTER IX THEOREM FOR ORTHONOMIC SYSTEMS
RIQUIER'S EXISTENCE
98. In Chapter X, we shall extend some of the main results the preceding chapters to systems of algebraic partial shall find it necessary to use an differential equations. of
We
We
important existence theorem due to Riquier. develop this existence theorem now, following, in some respects, the concise exposition of Riquier' s work given by J. M. Thomas.*
For the to a finite
proof, in Chapter X, that every system number of irreducible systems, only
present chapter, which can be read immediately,
is
equivalent
106 of the is
necessary.
MONOMIALS
xm We deal with ra independent variables, x\ l x a an where is meant By monomial, expression x\ the ik are non-negative integers. If a yft, with a, ft, y a then a of is called Given two distinct ft. monomials, multiple 99.
,
,
.
l
=
,
monomials, /y.*l
.
.
.
*!
the
~> l
m
^m
rfJ\
.
*!
>
.
.
/v*^m
^m
>
said to be higher or lower than the second according non-zero difference jk is positive or is negative. due to Riquier, is used only in following theorem,
first is
as the
The
4
first
Chapter X.
THEOREM: Let (1)
be
i,
an
which
2,
?
?>
of monomials. Then there a multiple of some aj with j
infinite sequence is
* Annals of Mathematics, vol.30 (1929), found in this paper.
135
p. 285.
is
an
References will be
10
ALGEBRAIC DIFFERENTIAL EQUATIONS
136
A
Let be one of those t for which the exponent of Xi a minimum. Consider the monomials which come after fa in (1). Let fa be a monomial of this class whose degree in Xi does not exceed that of any other monomial of the class. Of the monomials which follow fa let fa be one of minimum is
,
degree in
xlt
we form an
Continuing,
monomials
whose degrees from
(2),
decrease.
We
in a* are non-decreasing.
extract similarly,
a sequence in which the degrees in x2 do not We arrive finally at an infinite subsequence of (1)
which each monomial
in
-
A, A, A,
(2)
sequence of
infinite
is
a multiple of
all
which precede
it.
DISSECTION OF A TAYLOR SERIES
Let
100.
be the Taylor expansion at
a function u
of
=
Xi
(4)
xif
of
Let [] be any given monomials.
We
finite
xm
,
i
0,
analytic
at the
and non-vacuous
are going to separate
(3),
=
1,
point
,
m
(4).
set of distinct
with respect to [],
into a set of components.
Let a be the greatest exponent of xv
in the set [].
We
write
u
(5)
fori
where,
all
i
terms in (3) in which the
As
to
af/a
,
it
contains
all
terms divisible by o^. Then fv fa _ 1 are series in involves also while XL* #2, Xm, fa We define sets of monomials [a]*, A 0, -, a, as follows. If [] contains monomials in which the exponent of XL does ,
,
*
We
=
consider every combination
coefficients if necessary.
ti,
,
im to occur in (3), using zero
IX.
RIQUIER'S EXISTENCE THEOREM
137
not exceed A, then []A is to consist of all such monomials If there are no such monomials, then [a]^ is to be in []. -, xm obtained unity. Let [0]*. be the set of monomials in #2
=
,
1 in [a]*. We now give to each /A, with by putting x^ respect to #2 the treatment accorded to u above, with respect For A
,
b depends upon A, the/ju with i<.b involving xs ,--,xm For A while /A6 involves also #2 at involves o?i. a, each That is, in the dissection of /a we treat like x8 -, xm
where
=
.
,
/
-
^
,
,
.
We use operate on each f^ with respect to xs a set of monomials fy]^, where, if f/8]j, has monomials of degree not exceeding ft in x%, [y]^ is obtained by putting 1 in all such monomials, and where, otherwise, #2 [y]^ We now
.
=
is
unity.
we
Continuing,
find
an expression for u,
the summation extending over a finite number of terms. Example: Let u be a function of x, y, z. Let [] be xz*j
For Xj we find w /o (y,
=
We
now
treat
xy,
x*yz.
+ a?/i to, ^) + ^ /2 (xj 2
)
each/* with respect to y, the
y, z). set of
monomials
being that indicated below;
y,z
g, y,yz.
Hence to, ^) ,
(x, y, z)
== /,
(x, z)
+ y/M
(a?,
y
,
*)
.
10*
ALGEBRAIC DIFFERENTIAL EQUATIONS
138
The
final step is
=/ooo(y,
/oo(2/, z)
1;
*)
= /100 + Z/101 + * /102 = Aw(y) + */m + *Vii2 = /2oo + 2/201 + /202 = f2io(x,y)+zf2ii(x,y) Jrz f2i2(x,y,z) 2
/10
*
(*)
(*)
/a (y,
(y, *)
(y)
z)
2 ;
1,
2
/2o (# s)
(#)
,
(a;
(a?)
z*\
z)
,
2
fu(x,y,z)
Thus the dissection
=
w
u
of
l,
is
+ ^/loo + #/ioi + #* /io2 + #y/no (y) + xyzfm (y) + xyz*fn* (y, z) + ^/aoo 0*0 + ^ ^/2oi + # ^ /202 (, ^) 2
/ooo (y
,
*)
0?)
2
2
2
(a?)
=
101. Consider any monomial a o x^ in [a] and is a multiple of in u which monomial the ft expansion any of Of course, ft appears in one and in only one of the 1
.
terms
Xl
l
'
'
Xm
of
tiple
second
the
in '
fi
a
=
r.
m
For
the result be
m
^6 =
'
-i "
1
.
We
member of S ^ a ^ P r0ve
true
1
Suppose
Among
first
ii
f
t
1
.
xl
'
it
'
'
xm
shall
^S
a
with in (5).
Let it
prove
in
mU ^"
holds.
certainly
We
app'ear
for
of w,
i^jr Then
l
ft/x[
free of XL
is
.
the monomials used in the dissection of f^ will be
xrj r and
#2*
that
l
^ at
result
l
that in the resolution (5)
first
appears in a term x[
ft
,
m=r
for
observe
this
Let
(7).
1
ft/x^
there are only r in a term cfi^..
1 >i
r
will
be a multiple of
As
.
l
variables involved now, ft/x[ will appear
in the dissection (7) of l
jr
xjrr
ocfy
/^ with
divisible
r
Thus x[ xr is divisible by xr #2* a. Then //# is contained in /a Suppose now that ^ in the used the dissection of fa will be monomials Among
by
xi*
l
.
.
=
'
'
'
xir
*
Now
.
the formal scheme in (7) of the dissection
fa can be obtained by taking a function g of x2 XT, dissecting g with respect to the monomials associated with /a of
,
,
and then adjoining a?i to the variables in the series yielded l xrr in the dissections, That is, the monomials #**
by g.
IX.
RIQUIER'S EXISTENCE THEOREM
139
analogous to (7), of /a and g, will be the same. Let y result I Then y is found in the dissection from ft on putting x^
= l
.
xjrr The same xr divisible by xi* g with an x a would therefore be true for ft/x in the dissection of /a of
r
-
.
.
This completes the proof. It follows that
every monomial in []
is
an #*
1
x% in (7). x% in (7)
1 02. The set of monomials consisting of all a?* which are multiples of monomials in [a] will be called the extended set arising from [a] The set of monomials a?* x% in (7) not in the extended set will be called the set com1
1
.
plementary to []. If [a] is identical with the extended set arising from [], then [] will be called complete. Consider a set [a] which is not complete. We shall prove that it is possible to form a complete set by adjoining to [a]
multiples of monomials in []. Let p be the maximum of all exponents in all monomials in
[]. Then, in (7), no 4 exceeds p. Let []' be the extended set arising from [].
Then
if
a proper subset of its extended we can never get more than (p l) m in (7), this process of taking extended
not complete, it set [a]" ( 101). Since
[]'
is
is
+
monomials x[ x% sets must bring us eventually to a complete set. 103. In (7), the variables in an /^...^ will be called the l
of the
multipliers
Let
and
x%,
all
other
l
will
l
if
=
ft
l
be called non-multipliers of x[ x%. Of x% has no multipliers. /^...^ is a constant, x[
variables course,
corresponding x[
a?*
1
a?!*
1
Tfl
be a monomial in the extended set
Let xt be a non-multiplier of ft. Then some monomial in [], is the product of a monomial y in the extended set by multipliers of y ( 101). a. We shall prove that y is higher than ft. Let y a^
arising from /tftf/c,
[a].
as a multiple of
=
If
.71
<
*'i ,
Xi cannot be a multiplier for y since
not the
maximum
Hence j
l
ji
=
ii.
>
%.
of the degrees in xi of the It
When we
;'i
is
certainly
monomials
in []. remains to examine the case in which
dissect /^,
we
find that
if
jt< i2 x2 ,
can-
ALGEBRAIC DIFFERENTIAL EQUATIONS
140
>
Hence fa not be a multiplier for xt* i* and we ofy. have to study the case in which j* i* Continuing, we see that y is not lower than ft so that, since y ^ ft, y is -
=
higher than
ft.
We
104.
.
associate with
every monomial x^
xfa the
differential operator
Then the product
two operators corresponds
of
of the corresponding
to the product
monomials.
Consider any monomial
ft
= x[
l
x%
in (7).
Let the
corresponding differentiation be performed upon u, and after the differentiation, let the non-multipliers of ft be given zero
Every term in the expansion of u which is not by ft will disappear during the differentiation. Any term divisible by ft whose quotient by ft contains nonvalues.
divisible
ft will disappear when the non-multipliers are made zero. Hence the above operation gives identical results when applied to u and to ftf^.-.i^
multipliers of
105.
We
study,
for
its
instructive
There
xl
l
'
rather than of
will
"
each
value,
equations. application, a special system be as many equations as there are monomials xm* n (') (A set l] is supposed to be given.) With
for purposes of
ft
where
= x[
l
x%
-
in (7),
an
we
associate an equation
assigned function of the multipliers of /, analytic for small values of the multipliers.* shall show that there is one and only one function u,
^...^
We
is
arbitrarily
= 0, =
m, such that each equation the non-multipliers of the associated ft are zero, for small values of the multipliers. We are not are finding an actual solution of the system (8). finding analytic for Xi
(8) is satisfied,
i
1,
-
,
when
We
* If
ft
=
1,
the
first
member
of (8) is u.
IX.
RIQUIER'S EXISTENCE THEOREM
141
a u which satisfies each equation on a spread associated with that equation. Consider
any particular equation
member be to #1
#2
to
Let the second
(8).
times in succession with respect integrated to Xi then 4 times in succession with respect i
from from
,
to x*
,
and so
where /^...^
is
result will be a function
im 4 x ^m /V*m
il x ^l
^ ' (9)
The
on.
a function of the multipliers of
analytic
ft,
The function (9) satisfies its associated equation in (8). Let u be the sum identically of all functions (9) obtained from (8). The expression for u as a sum gives the dissection (7) of u relative to [a]. Then, by 104, u satisfies each equation (8) on the spread associated with that equation. If u is a second solution of for small values of its variables.
the problem,
it
is
u relative Hence u is unique. of HI
from
seen, to
104, that in the dissection
the
[],
series fi^.-im are
all
zero.
MARKS Let
1 06.
Biquier
j/i,
effects
,
an
y n be analytic functions of the yt and
of
xi9
their
ordering
,
xm
.
partial
derivatives in the following way.
Let
s
s
(10) in
We
be any positive integer.
Xi any ordered set of
non-negative integers Mil,
first
(11)
vu,
taking care that y{ and
j/j
sets (11).
His
,
With each integer, UH, is unity. set of ordered non-negative integers any
which the
associate
associate with each
The ^th integer
and the jth integer
-, with
i
^j
do not have identical
in (10) is called the ^'th
in (11), the ^'th
w
fc
= ^ a
t
n
we
t?i,
mark
If
(12)
y*,
m
\-k
s-y,'
of
yt
.
mark
of xt,
ALGEBRAIC DIFFERENTIAL EQUATIONS
142
we
define the Jth -j-
~T"
w2
1
be v
to
s
,
,
*
Let w l and Let the marks of
of the derivatives of all
all
any two
and
of w, j
Km Umj
Consider
be
mark
these
of
derivatives.
j/
r
be
identical.
We
Suppose that the two sets of marks are not shall say that wi is higher than w2 or is
lower than
w2
according as the
respectively.
first
non-zero difference
en
bi
If the two sets of marks are negative. identical, no relation of order is established between u\ and w 2 is
or
positive
is
.
If
wi
higher than
tv%, dwi/dxi is higher than dw*/dxi. always higher than w.
is
dw/dxi is When the marks in (10) and (11) are such that a difference order exists between any two distinct derivatives, the
Also, in
derivatives of the
yi are said
Suppose that the ordering
how
to adjoin
new marks,
to be completely ordered.
is
We
not complete.
after UiS
and
shall
show
so as to effect
Vis,
a complete ordering. Clearly, the adjunction of such marks will not disturb any order relationships which
new
may
already exist.
Let
m
additional
marks be assigned, as
in the following
table:
xm
xi X2
s+l s
+
s
+m
...
1
2
...
1
..
0, 0,
1,
2/1
yn
2/2
00 00 00
...
0,
...
0,
...
0.
wi and w a be two derivatives with the same set of s m marks. The (s i)th mark of wi or w2 i 1 m, is the number of differentiations with respect to xt in wi
Now,
let
+
+
or w*.
Hence the same
as in w*. it
follows *
Each
y.
From the now that
,
,
differentiations are effected in w^
definition of the
marks
the functions of which
wiU be considered
=
,
and 2 and w% are
of IV L
w
as a derivative of zero order of
ie/-
itself.
,
IX.
RIQUIER'S EXISTENCE THEOREM
have the same sets
derivatives
new
identical, so that the
In
everything
,
,
;
,
complete. shall deal
be variables.
n
,
is
we
which follows,
complete orderings. 107. Let t m 1 with w, in (12) the monomial
Thus n\ and
(11).
ordering
w
2
are
only with
We
associate
l
j ?^* r number of distinct derivatives of the y Let the monomial associated above with ivit i Let g be any positive number. We 1, ...,<, be a*. shall show how to assign, to the $, &, real values, not less
Let
ivi
,
,
w;*
be any
finite
.
=
t
than unity, in such a
way
if
that,
wi
is
have, for the assigned values, *>#/. introduce s new variables z,
We
higher than Wj,
,
5
-
we
With each &
u
" where the w.. are the zs monomial z^ /'' marks of #.. With each f we associate /" where i o * Then each f goes over into the v are the marks of y r f/
we
associate the
-
-
.
t
.
= z{
a* zs
the jth mark of w.. aj prove that we can attribute to real values not less than unity in such a way that
a monomial
fti
l
-
-
with
It will evidently suffice to
the
Zi
if Wi is higher than Wj. *Let r be the maximum of the degrees (total) of the A. Let k be any positive number, greater than unity and greater than g. We put
fti>gfy
(13)
Zi
Then,
=
if
with an fti
> \
&/i,
we
have, for (13), Zh
-^
ORTHONOMIC SYSTEMS 108. Let 2/1, a?m, j/n be unknown functions otxi, whose derivatives have been completely ordered by marks. ,
,
We
consider a finite system
of differential equations,
ALGEBRAIC DIFFERENTIAL EQUATIONS
144
--
84 ...8. m A
where
xm and of in each equation, g is a function of Xi, , a certain number of derivatives of the yi y every derivative in g being lower than the first member of the equation;
(a)
members of any two equations are distinct; a first member of some equation, no derivative of if w appears in the second- member of any equation; the functions g are all analytic at some point in the space
(b) the first
w
(c)
(d)
of
We
is
arguments involved in do not assume that every
the
Riquier
calls
all
of them.*
yi appears in a first member. such a system of equations orthonomic.
The derivatives of the yi which are derivatives of members in the orthonomic system are called principal All other derivatives are
rivatives.
called parametric
first
dederi-
vatives.
109. Given an orthonomic system, a, we shall show how an orthonomic system with the same solutions, in
to obtain
which, for each yi appearing in the first members, the monomials corresponding to those first members which are derivatives of yi form a complete set
(
102).
Let equations be adjoined to (14), by differentiating the equations in (14), so that, for each yt which occurs in some first
member, the monomials corresponding
set of first this
members
can be done.
to
the
enlarged
constitute a
We
102, complete set. By obtain thus a system cr, of equations.
members in 0i may be obtainable from more than one of the first members in cr. In that case, we use any one of the first members in <x which is available. Consider any one of the equations in
first
its first
member, and v the highest derivative
member. the
ad,
If first
we
differentiate
member becomes div/dx*.
*
in the second
the equation with respect to
The highest
deri-
Thus, in (d), derivatives not effectively present in a g may be reThis does not conflict with (a), in which garded as arguments in that g the arguments considered are supposed to be effectively present. .
IX.
RIQUIER'S EXISTENCE THEOREM
new second member
vative in the
dw/dx
lower than
t
will
145
be dv/dxi, which
is
106).
(
It is clear on this basis, that oi satisfies condition (a).
We
the second of
tf
now
attend
space of the
to (c).
arguments
members are
be an open region in the members in o* in which
(
We
analytic.
consider those solutions
which the indicated arguments
for
The second members the second members in be
Let
in the second
polynomials
in
in oi
may
lie in
(.
involve derivatives not in
The second members the new derivatives, with a.
in
oi
will
coefficients
analytic in (.
w
Let
be the highest derivative present in a second member is a derivative of a first member in oi. Then
in oi
which
iv is
not present in any second
member
in
so
that
it
appears rationally and integrally in the second members in oi Let w be a derivative of v, the first member of the equation .
=g
in oi Then w can be replaced, in the second members in oi, by its expression obtained on differentiating g. obtain thus a system ov with the same solutions as
v
.
We
members as *i. The The derivatives system higher than w which appear in the second members in o*2 also appear in the second members in oi. Hence, if MI, present in the second members in o"2 is a derivative of a first member in is lower than w. We treat w^ as w was treated. 0*2 then wi (or
oi
o),
o-2
and with the same
first
satisfies condition (a).
,
Since there cannot be an infinite sequence of derivatives each lower than the preceding one, we must arrive, in a finite
number of steps, at a system T, with the same solutions as cr, which satisfies (a), (b), (c), and which has complete sets of monomials corresponding to its first members. The second members in T will be polynomials in any derivatives not present in the second members of o-. Hence assumption (d) is satisfied for and for any values of the new derivatives. Thus r is orthonomic and has the same solutions (
as o\* *
in
With .
.the values of the
arguments in the second members in o lying
ALGEBRAIC DIFFERENTIAL EQUATIONS
146
Of course, whether we employ o" or r, we get the same set of principal derivatives and the same parametric derivatives.
no. We
whose
first
109, yield complete sets of monomials.
We
consider an
members, as in
orthonomic system,
seek solutions of
are going to
cr,
which, with no loss of generality, i
=
1
,
,
m
analytic
may
or,
at
some
point,
be taken as x%
= 0,
.
Consider any
yi.
Let numerical values be assigned
to the
origin, with the sole parametric conditions that the second members in a are analytic for the
derivatives
at the
of yi,
values given to the derivatives in them and that the series
where the a are the values of the parametric derivatives, the subscripts indicating the type of differentiation, converges The series (15) is called in a neighborhood of the origin. If yi does not appear in the initial determination of y%. a first member, (15) is a complete Taylor series.
In
what
given
for
calculating
follows,
we suppose an
We
initial
determination to be
then develop a process for yi. the values of the principal derivatives at the
each
shall
There will result analytic functions yi which satisfy each equation of a on the spread obtained by equating to
origin.
zero the non-multipliers of the monomial corresponding to the member. Later we shall obtain a condition for the yi
first
to give
an actual solution of
cr.
In the dissection (7) of each yi which we shall obtain,* those terms whose monomials are multiples of monomials in the complementary set will constitute the initial determination of yi.
Thus the
initial
determination of each yi
is
a linear
combination of a certain number of arbitrary functions, with monomials for coefficients, the variables in the arbitrary functions being specified. * This
dissection
sponding to
yi.
is
This description of the degree of
based on the complete set of monomials corre-
IX.
RIQUIER'S EXISTENCE THEOREM
147
generality of the solution of a system of equations of the most important aspects of Riquier's work.
We
replace
each
yi
which does not
member in er by an arbitrarily selected Then a becomes an orthonomic system
figure
initial
in
one
is
any
first
determination.
in the remaining yi, with the same principal derivatives as before for the remaining yt. On this basis, we assume, with no loss of
generality, that every yt figures in a first member. in. use the symbol d to represent differential operators.
We
Any in 0, in
er
principal
dyiy which
derivative,
is
not a
first
member
can be obtained from one and only one first member by differentiation with respect to multipliers of the
monomial corresponding to that first member. This is because the first members yield complete sets. We have thus a unique expression for dyi, d yi
(16)
=
ff,
where the derivatives
in g are lower than Sy it system obtained by adjoining all equations (16) Let p be any non-negative integer. will be called r.
The to
(5
infinite
The systems of equations in T whose first members have p for first mark will be called rp Since the first mark of a derivative is the sum of the order of the derivative and of the first mark of the function differentiated, each TP has only a finite number of equations. .
b the maximum, of the first marks For the values assigned, in 110,
Let a be the minimum, and in the first
members
in
er.
-, i^ parametric derivatives, the equations ra ,ra +i, determine uniquely the values at the origin of the principal
to the
derivatives
whose
first
mark does not exceed
&.
In short, the
lowest such derivative has an equation which determines it in terms of parametric derivatives; the principal derivative next in ascending order is determined in terms of parametric deri-
vatives, and, perhaps, the first principal derivative,
We
subject the
unknowns
and so
yj to the transformation
on.
ALGEBRAIC DIFFERENTIAL EQUATIONS
148
where
y; is the chosen initial determination of yj
the c are
and where
the principal derivatives at the origin of not exceeding 6, found as above.
yj,
of
mark Then a goes over into a system a' in the yj. In the new system, we transpose the known terms in the first members (these come from the known terms in (17)) to the right. The new system will be orthonomic in the j/}, with the same monomials for its first members as in o*. The second members will be analytic when each Xi and each parametric derivative first
is
small.
The system
analogous to r for
for
which are identically zero, the principal derivatives at the origin, of first mark not exceeding 6, will be determined as f
zero by r aj
On
.
.,
r'b
.
account, we limit ourselves, without loss of to the search of solutions y^, , yn of
this
generality,
with
-
,
determinations identically zero, assuming that r b yields zero values at the origin for the system r a the principal derivatives whose first marks do not exceed b. initial
,
,
In the second members in
112.
374-1,
no derivatives appear
whose first marks exceed 6 + 1. Those derivatives whose 1 enter linearly, because they come from marks are 6
+
first
the differentiation of derivatives of
first
mark
We
6 in r&.
denote by d k yi the second member of (12). every equation in ra+i is of the form
Sty*
(18)
where the
dj
yp are of
=
2piajpdjyp
first
mark
6
+1
involve the x\ and derivatives whose
In (18),
which
is
we
qi<*
and where the p and q marks are b or less.
first
+
1 consider every derivative of first mark 6 member. the second in to be di y tt present
lower than
If necessary,
+
Then
we
take
p^p
=
.
Consider any diya in (18). Suppose that there is a ft such 1 which are lower that yp has derivatives of first mark 6
+
than
<Jty
For every such
ft,
we
let rtap
represent the
IX.
number
RIQUIER'S EXISTENCE THEOREM
of derivatives of yp, of first
lower than
<J z
i/a.
For every other
mark
fi,
we
&
149
+ 1,
let r ta/j
which are 1, and
=
we suppose
that a single derivative of yp of first mark appears in the second member of (18), with a zero We can thus not continue to say that every coefficient. derivative in the second member of (18) is lower than the &
+1
first
member, but no
difficulty will
arise
out of this;
only
a question of language is involved. Let r be the maximum of the n ap.
The p and arguments.
q in (18) are analytic for small values of their series of powers
Let the p and q be expanded as
of their arguments.
>
Let
be such that each of the above series converges which all exceed e in modulus.
for values of its arguments
Let h
>
be such that each
p and each
q has a modulus
than h when the arguments do not exceed Let A be any positive number less than l/n.
less
c.
107, we determine positive numbers Following less than unity such that, if 8i y a and dj yp are of with diycc higher than djyp, we have
?*,
&, not
first
mark
>
In what follows, function m.
we
associate with each yt a
new unknown
Let
first
2
u n whose ranges over all derivatives of i^ is b mark does not exceed (8iU supposed to have the
where
,
same marks as <^y).
We
consider the system of equations
(20) ,
,
ALGEBRAIC DIFFERENTIAL EQUATIONS
150
which has the general form form of the p and q.
The
of (18), with
alterations of the
function
h
is
p and
a majorant for every
same
least unity, the
every
q.
true of h/(l
is
As each
?
is
at
e).
Thus, in virtue of (19), wherever a dj-yp is lower than diy n in an equation in (18), the coefficient of djiip in the corresponding equation of (20) will be a majorant for the coeffiIn the exceptional case where a djyp is not and thus has a zero coefficient, the corre-
cient of 8jyp.
lower thaji
<5i?/
sponding coefficient in (20) is certainly a majorant. Evidently the terms in (20) which correspond to the q in (18) are majorants of the q.
We
113.
each
show
shall
a solution in which
a function of
in is
(21)
?l
Consider, in (20), marks are I
+
first is
that (20) has
*l+"-
all
1
derivatives of a particular U K whose first mark of any such derivative
The
.
the order (total) of the derivative, plus the first mark of w. all of the derivatives of u a which are of first mark
Hence
&+1
are of the
same what
order, say
#.
Let the ?(, in follows, represent functions of (21). Put ua ?4 and let Uai be the (i Oth derivative of u'a with respect to (21). The)
=
+
V"
8
lm
8^..-a^ When gcc,
(22)
=
the
?c
_ ~
g
.
.
.
^^
are functions of (21), Q becomes a function ?' of the un of order less than
and of the derivatives 1
,
.,
u*
'
$<* t M -'
7t
1
of (21)
ua
+
n.
=
Equations (20) reduce to
RIQUIER'S EXISTENCE THEOREM
IX.
There
be n equations
will
in
one
(22),
which a given ?< the same member yield equation (22). in
equations in (20)
each
for
appen
We
151
vs
in
.
All
the
first
write (22) as
n f
/t)o\
i
f [
*
ffa
When each
(21)
0a
T
'
^sr*
i
p^
when
zero and
is
2
fffl
the ui,
i
= 0,
-,
y
tt
1
,
for
are given zero values, the determinant of (22) with respect to the n'ag a is ,
1
/
,
This determinant
is
,
I
A, ....
1
i,
A
A
f* ,
"~
j
A
.
,
not zero.
A
I
In short, the equations
(24) Zn
=
CH
imply
so that the determinant cannot vanish for
Then the u KQoL can be expressed as functions of the other quantities in (23), analytic when the arguments are small. the existence theorem for ordinary differential equations,
By
(23) has a solution with the is
The functions
zero.
u
tt i
zero,
for
?'<#, when
in this solution will
(21)
be analytic for
(21) small.
114. u'ai with
We
shall prove that, in the solution just found, all
i
;>
=
1
g a are positive
for (21) zero.
For
(21) zero,
we have
*
For
i
,
subtract each equation from the
substitute the results into the
first.
first,
in succession,
and
ALGEBRAIC DIFFERENTIAL EQUATIONS
152
Referring to (24), we see that, since A
.
Differentiating (23),
where the k
l(
we
find,
for (21) zero,
Again, the solution consists of Continuing, we obtain our result.
are positive.
positive numbers.
What precedes shows that (20) has a solution, analytic at the origin, with every derivative of first mark less than l-\-l equal to zero and every other derivative positive, at the origin.
We now
115.
employed,
in
the principal
we determine
With the procedure return to the system a. at the origin, of the for determination, 111,
derivatives
of
first
mark not greater than
ft,
principal derivatives at the can origin. ascend, step by step, through all the principal 111 has only a finite number derivatives, because each TP in
the values of
all
We
of equations.
We
power series for each yi. We power series converge for small
obtain thus a complete
are going to prove that these
values of the
Let
dtt/u
;r;.
AVe shall prove that derivative at the origin does not exceed
be any principal derivative.
the modulus of this
the value at the origin found for ^?e
For derivatives all
than
mark
less
than
derivatives lower than
ft.
The equation
114.
in b
+1
some diyy
,
this is certainly
Let the result hold
those derivatives have zero values.
true;
for
of first
of first
mark greater or
in (18), in r for d^j y is either
found by differentiating some equation in corresponding equation for <5/?< ; ,, which obtained from (20) by differentiation.
(1 8).
is
is
Consider the
either in (20),
or
AVe shall consider the expressions for diy y and dtn y as power series in the xi and in the derivatives in terms of
which 6,y y and d n are expressed. y AVe see that, for every term in the series for t
is
a dominating term in the series for
di*t y
.
<J,-j/
What
y
,
is
there
more,
RIQUIER'S EXISTENCE THEOREM
IX.
the
series for
itself,
ditty
may have
other terms,
or even higher derivatives.
This
is
153
involving d^iy because of the
112.* exceptional terms in (20), introduced in Each term in di u r which has a corresponding term in
dj y y modulus of that term at the origin, for such terms involve only lower derivatives than dnjy or Terms in d uy which have no corresponding terms in ditty. is
at least as great as the
t
yy are zero or positive at the origin. They will be positive if they involve no #$, and contain only derivatives of first mark at least b~\-l ( 114). This proves that the value di
determined for each 6iy a by z has a modulus not greater than the value at the origin of dn< a Thus the series obtained for the /// converge in a neighborhood of the origin. .
116.
We
shall
now
what extent the
see to
tions ytj just obtained, are solutions of
analytic func-
a*.
=
all
Consider any equation dyi y in tf. This equation, and it obtained from equations by differentiation with respect
monomial corresponding to the lirst member, are satisfied, at the origin, by the derivatives of 2A Un at the origin. Hence, if we substitute ?A ,-,*/ into dy #, x n which vanishes at the we obtain a function k of x\ origin, together with its derivatives with respect to the above to multipliers of the
,
,
{
,
multipliers.
occur.
Then
Hence y
Thus, in the expansion of k, only non-multipliers are zero. Jc vanishes when the non-multipliers G on the each spread y n satisfy equation of -
,
,
obtained by equating to
the first
,
to
member of
117. Let us return
zero the non-multipliers corresponding the equation.
now
to the
most general orthonomic
system tf whose first members give complete sets of monomials. We do not suppose that every y\ appears in some first member.
We
consider any point xt
=
en,
to obvious conditions of analyticity.
i
=
1,
,
m, subject
Let any values be given
to the parametric derivatives of the yi at #1, a m so as Then the principal to yield convergent initial determinations. ,
*
In our present language, all derivatives in a second are lower than the first member.
member
?
in (18)
ALGEBRAIC DIFFERENTIAL EQUATIONS
154
by a in such a way as -,?/>* which satisfy each
derivatives are determined uniquely
yield analytic functions 2/1, each equation in cr on the spread obtained by equating to to the first member the of non-multiplier ^-corresponding equation. to
m
This
an immediate consequence of the preceding sections.
is
PASSIVE OUTHONOMIC SYSTEMS 118. Let
be an orthonomic system, described as in the Let the equations in # be listed so that
rf
preceding section.
members form an ascending sequence, and
their first
be written
If
=
vt
(25) i'i
is
8yj
establishes
we
#,
convention of
be the marks of the
in vi.
the
By
to
Vi,
=
the s marks of
the the
Thus,
vi,
them
1,
,
8ijj.
according
derivatives of
f.
This
to the tv,
we
the marks of dvi will
derivative
highest
monomial corresponding sponding to 8yj.
of
all
106.
as in
vt
among
To
106.
marks
attribute
attribute to
order relations
*
0,
let
we mean
in
dn.
By
the
the monomial corre-
We
shall refer to dyj as the first term term of a derivative of vi, we shall mean
first
the corresponding derivative of dyj. Consider a v whose corresponding monomial, Let xi be such a non-multiplier. multipliers.
,
has non-
By
103,
product of a ft, in the same complete set as and higher than a, by multipliers of ft. Hence, there is a vpj higher than v, such that some 8v p has the same first term Xi<* is the
as dv/dxi.
Then, in the expression
_*
(26) all
derivatives
effectively present
are lower than
the
first
term of dv/dxi. It
is
clear that
derivatives principal
as
w
a polynomial in such principal Let w be the highest such Then w is the first term of some ex-
(26)
may
derivative.
is
is
involve.
where dvq
pression 8v qy Vq so that
it
is
obtained from
lower than dv/dxi. it
by
We
choose
differentiation with respect
RIQUIER'S EXISTENCE THEOREM
IX.
to
Let then, identically,
v q unique.
(27)
w
=
where the derivatives
in
k are
w
in (26)
Thi* makes
corresponding monomial.
the
of
multipliers
loo
+ k,
all
lower than
ic.
We
replace expression in (27) and find, identically.
its
by
dv q
=
&Vp
-7j^7
+ hi(8v
-
q
-
),
,
hi is a polynomial in 8v q whose coefficients involve no principal derivative as high as iv. Let u\ be the highest We give it the treatment accorded principal derivative in hi
where
.
w and
to
where
find
is
/^
a
-~~
(28)
w hich T
in
polynomial in dv
(1 ,
<Jtv-
we
Continuing,
find
manner, an identity
in a unique
=
cH^
the coefficients in h involve only parametric deriwrite (28) in the form
We now
vatives.
~
(29)
where
/*
expression
the term
is
in
the
of
zero
degree in h. derivatives
parametric
Then
alone.
i*>
an
is
The
ex-
8vz are replaced by 0. pression y vanishes when dv q 0. It is clear that, for any solution of
,
The
totality of equations
of
for
cr
^ = 0,
which the monomial corresponding
has non-multipliers,
all
If
equations
to the first
member
every expression ^
is
er.
identically zero, the system
cr
said to be passive.
is
We 2/i,
in
all
non-multipliers being used, are called
the integr ability conditions for
119-
=
obtained from
shall
prove
that,
if
is
passive,
the
n
functions
117, which satisfy each equation yn, described in on a certain spread, constitute an actual solution of #. >
o-
What we have in (25)
to
show
is,
vanishes identically.
that for these functions, every
vt
ALGEBRAIC DIFFERENTIAL EQUATIONS
156
When
the yj
m
a function
=
lit
0.
above are substituted into
of y\
,
,
m
Otherwise,
xm
we
r/,
obtain
n
has no non-multipliers, vanishes when the non-multipliers If
.
monomial corresponding* to ?v are equated to their /. in If, (29), where /* is now identically zero, the parametric derivatives in / are replaced by their expressions as functions of the Xij found from the y/ (29) becomes a system y of differential equations in the unknowns vi. Since (29) consisted
of the
z ,
of
identities,
=
t'i
now
each
satisfied
is
y
replacements,
by
/=!,,.
id,
We to
before these
attribute
to
each
an
jr t
an additional mark
Vi
mark
additional
With
i.
t
0,
and
this change, the
will be completely ordered and the first each equation in y will be higher than every derivative in the second member.
derivatives of the
member
-
?; t
in
second members in y contain derivatives of the first members, we can get rid of such derivatives, step by step. Then y goes over into an orthonomic system i/S with the If the
same first members as y. For our purposes, it is unnecessary to adjoin new equations 109. to iff as in Consider any unknown VL which appears The derivatives of r in the first members in a first member. t
will
be taken with respect to certain variables
(30)
Xa
The variables
m
which
(30)
'
*d-
'
',
,
when equated
to their a/, give a spread on
vanishes.
The parametric
derivatives
of
vt
will
be the
derivatives
taken with respect to the variables not in (30). For the corresponding m, each of these parametric derivatives is zero.
Now we know
that,
derivatives, there is at i
=
i
. J
.
. ^
t
is
given
values
most one solution
a solution of
derivatives vanish.
is
for
Hence
m
V
= 0,
the
of of
f or
which
i
1,
parametric
But
*/>.
all ,
vi
0,
parametric
t.
This proves that, given a passive orthonomic system, there one and only one solution of the system for any given initial
determinations.
CHAPTER X SYSTEMS OF ALGEBRAIC PARTIAL DIFFERENTIAL EQUATIONS DECOMPOSITION OF A SYSTEM INTO IRREDUCIBLE SYSTEMS 1
20.
We
unknown functions, ?/1? Definitions xi,-'-,x m
consider n
,
yn
,
of
m
will
independent variables, usually be as for the case of one independent variable, and will be .
given, formally,
We in
only
assume marks
such a
way
the y it By a form,
as
we
when
there is some necessity for have been assigned to the xi and
to
it.
yt
order completely the derivatives of
to
mean a polynomial
and any number of their partial derivatives, with coefficients which are functions of the Xi, meromorphic at each point of a given
open region
21
shall
in the space of the xi.
in the yi
By
a
field,
we
shall
mean a set of functions meromorphic at each point of 21. the set being closed with respect to rational operations and assume a field & to be given in partial differentiation. advance. Where the contrary is not stated, the coefficients
We
in a
form will belong to
121.
By
unknowns, we in
r.
the leader of a form shall
mean
the
A. Let
A
which actually involves
highest
derivative
present
A l and A2 be two forms which actually involve unknowns. A* has a higher leader than A i7 then As will be said to be of higher rank than AI. If A and A 9 have the same common leader exceeds and A the if the of in 2 leader, degree that of Ai then again, A 2 will be said to be of higher rank than Ai. A form which effectively involves unknowns will be said to be of higher rank than a form which does not.
If
,
157
ALGEBRAIC DIFFERENTIAL EQUATIONS
158
Two
forms, for which no difference in rank is created by precedes, will be said to be of the same rank.*
what The lemma
2 goes over immediately to the case of
of
several variables.
122.
If
A
the leader of AI.
A^
if
A 19 A t an ascending
will be called (a)
(b)
A$ contains no derivative (proper)
if
A ^
and
I
v
is
lower degree than AI in
of
set of forms
(1)
r =
A* will be said to be
involves unknowns,
AI
reduced with respect to AI of the leader of AI and
..-,
,
Ar
set if either
0, or
AI involves unknowns and, for j>i, Aj rank than At and reduced with respect to At.
r>l,
When for
(b) holds, the leader of
Aj
is
is
of higher
higher than that of
A,-
j>i.
Relative rank for ascending sets is defined exactly as in 3. If
2>
(I)
3
then
LEMMA.
(P 1
>CD 8
.
(2)
!,
an
We
prove the following lemma.
Let 0> 2
,
-..,
q
... ,
of ascending sets such that
infinite sequence
*
For
q large, the
These
first
first
forms in the
q
have the same rank.
forms will either be free of the unknowns, or We have only to
have the same leader, say p r consider the latter possibility, and may else
will
limit
ourselves to
q with q large has at least two forms. For q large, the second forms will have the same leader, Now p,2 is not a proper derivative of p^ As we say p saw above, p 2 is higher than p r We may confine ourselves
the case in which
.
now
to the case in
* It will
x,-
7
,
for q large, has at least three
be noticed that the above definitions of relative rank do not
specialize into those of of each
which
is
unity.
2.
This
is
due to the fact that the
first
mark
PARTIAL DIFFERENTIAL EQUATIONS
X.
159
Then, for q large, the third forms will all have the same leader, j?8 higher than p l and p 2 and not a derivative
forms.
,
of
either of them.
an
infinite
Thus,
our result holds, unless there
sequence Pi> P*>
>
is
P q '" >
derivatives which increase steadily in rank, no p being p i with i
a derivative of a
On
of the
the basis
a
of
above lemma, we define a basic
2 which contains non-zero of 2 of least rank.
system
forms,
to
set
be an
ascending set If AI in (1) involves unknowns, a form will be said to be reduced with respect to (1) it is reduced with respect
F
to Ai,
=
i
F
1,
-,
r.
be a system for which (1), with A t not free of the unknowns, is a basic set. Then no non-zero form of 2 can
2
Let
be reduced with respect to (1). If a non-zero form, reduced with respect to (1), is adjoined to 2, the basic sets of the resulting system are lower than (1). In this section, we deal with an ascending set A involves unknowns.
123. in
(1)
which
a form
If
has a leader, p,
The
the separant of G.
G
in
Let
and
Si
Ai in
of
be called the
will
We
shall
G
=
be,
initial of
respectively,
be
form
G/dp power of p 5
G. the separant and
1,
prove the following result. any form. There exist non-negative ,
initial
coefficients, is subtracted
the remainder,
integers,
$i,
when a suitable linear combination number of their derivatives, with forms
r, such that,
of the Ai and a certain jor
shall call the
(1).
Let ti, i
It
we
coefficient of the highest
R,
is
from
reduced with respect
to (1).
Let pi be the leader of Ai. We limit ourselves, as we may, to the case in which G involves derivatives, proper or
ALGEBRAIC DIFFERENTIAL EQUATIONS
160
improper, of the pi. Let the highest derivative in G which is a derivative of a pi be q and let q be a derivative of PJ. For the sake of uniqueness, if there are several possibilities
we
for;;,
use the largest
assume q higher than
pr
available.
;
To
our ideas,
fix
we
Then
.
=
SfG
CA$ + B
a derivative of Aj with q for leader, and where Because Aj and Sj involve no derivative q. B no derivative of a pi which is as than involves q, higher high as q. For uniqueness we take g as small as possible.
where Aj
B
If
we
is
free
is
of
B involves a derivative B the treatment
of a pi
which
higher than # r After a finite ,
D
which differs of steps we arrive at a unique form a linear combination of derivatives of the Ai from a form
number by
is
accorded to G.
give
D
The form
involves no derivative of a pt which
is
higher
than pr We then find a relation .
where
K
involve
p
is
K
reduced with respect to Ar The form may Aside from pr the only derivatives of the pi .
r.
,
K
are derivatives otpi> Such derivatives -, pr-i* present in Let q^ be the highest of them. are lower than p r the give Suppose that q^ is higher than jvi. .
treatment accorded to G. arrive at a unique form
by a
linear
Ai,
,
Ar-i.
and A?-*.
L
combination
The form
Aside from
In a
L
which
of
L
Ar is
p r and
are derivatives of pi, Pi in are lower than
,
finite
We
K
number
of steps,
differs
\
we
from some
and the derivatives of
reduced with respect to Ar jv-i, the derivatives of the jpr~2, and all such derivatives
PARTIAL DIFFERENTIAL EQUATIONS
X.
161
R
Continuing, we determine, in a unique, manner, a form as described in the statement of the lemma. the call
R
We
G
remainder of
with respect
to (1).
7-10 now goes over, without 124. The argument of to case of several the change, independent variables. We secure the lemma:
LEMMA.
Every
of forms in
infinite system
a finite subsystem whose manifold
to
in
we prove
13,
,
The decomposition is unique in the sense As an example, we consider the equation
=
z-(px-\-qy)-\-p* + q*
(3)
= dz/dx, q = dz/dy.
where p a;,
we
(rx
=
where r
yn
of
is
equivalent
14.
0,
Differentiating with respect
d*z/dx 2
,
+ sy) + 2(pr + qs) s
=
d*z/dxdy,
entiating (3) with respect to y,
we
t
=
This lemma
very different from, and
is
d*z/dy*.
Differ-
find
(8x+ty) + 2(ps + qt)
(5) *
y n has of the
find
(4)
theorem
,
the
THEOREM. Every system of forms in y^, a finite number of irreducible systems.
to
y\,
identical with that
system*
infinite
As
is
is
0.
not to be confused with, the
of Tresse for general (non- algebraic) systems of partial differential
equations. (Acta Mathematica, vol. 18, (1894), p. 4.) In using the implicit function theorem to solve his system for certain derivatives, Tresse has necessarily to confine himself to a portion of the manifold of his system. it is not easy to imagine systems other than linear systems for
In fact,
which Tresse's argument and result have a definite meaning. On this 129, may be basis, the above lemma, together with the theorem of regarded as an extension, to general algebraic systems, of Tresse's result, as applied to linear systems. Thus the relation between Tresse's theorem and our lemma
is
linear functions of
functions,
quite like that between the theorem that a system of n variables contains at most n 1 linearly independent
+
and Hilbert's theorem on th
existence of a finite basis for any The use of Riquier's theorem of
system of polynomials in n variables. 99 was suggested to us by what is contained in Tresse's work. is
the only
common
feature of the
two arguments.
This
ALGEBRAIC DIFFERENTIAL EQUATIONS
162
From
and
(4)
s
(rt
2
)(x
either
Thus,
(5),
we
obtain
= =
2p)
rt
s*
0;
(rt
or z
=
solution of (3) does not annul rt
As one can
system.
s
what
see from
-
s
2
(# 2
2 )
(y
+ ?/)/4.
Thus
.
- 2q) =
follows,
The
(3) is it
0.
latter
a reducible
is
equivalent
two irreducible systems.
to
BASIC SETS OF CLOSED IRREDUCIBLE SYSTEMS 125. Let
2
be a non-trivial closed irreducible system for
which
A l9 Ai,-.-,A r
(6)
is
a basic set.
vanishes
initial
A
,
solution of (6) for which no separant or be called a regular solution of (6).
will
Evidently such regular solutions exist. The remainder, with respect to (6), of any form of 2. is zero. Hence, every regular solution of (6) is a solution of 2. Furthermore, 2 consists of all
forms which vanish for the regular solutions of
We
m
more
(6).
a point use
represent by &, briefly by which the coefficients in (6) are analytic. the symbol fa] to designate any set of numerical values which ,
,
,
We
in 21 at
one
may choose
to associate with the derivatives appearing
The existence
in (6).
the existence of a set
but
,
fa]
which no separant
for
we
follows,
Let
of regular solutions of (6) guarantees
pi
the
xi
In what
vanishes.
initial
deal with such a set.
be the leader of Ai.
as an algebraic equation for pi of
which every At vanishes,
for
or
The equation
A
t
= 0,
treated
determines pi as a function and the derivatives lower than pi in A x the ,
,
function being analytic for xt close to & and for the derivatives lower than p t close to their values among the fa].
The value
of the function pi for the special arguments Let the above will be the value for p^ in fa] We can then solve expression for pi be substituted into A 2 for pz, expressing jp 2 as a function of the x% and of A%
stipulated
.
.
PARTIAL DIFFERENTIAL EQUATIONS
X.
p and p% appearing
the derivatives other than
We
the
substitute
4
solve
forms
3
=
for
t
for
expressions
ps
and continue, in
,
in
A
L
and p 2
pi this
and
into
manner,
A
t
.
A*,
for
all
We
in (6).
thus a set of expressions for the pi, each as an analytic function of the Xi and of the Pi being given derivatives other than p L9 write p r in (6). will
find
We
,
(7)
pi
=
i
gij
=
1,
,
considered as differential equations
If the equations (7) are
prove the
we
65,
see that
if
(6)
=
m
THEOREM: The orthonomic system in
We
form an orthonomic system.
for the yt, they will
As
r.
g\
shall
is passive.
considered as a set of
is
simple forms in the symbols for the derivatives, (6) will be a basic set of a prime system, -//.* The unconditioned will be those corresponding to the parametric unknowns in
A
derivatives in
We
(7).
=
W
(8)
the
bi
being integers.
form a simple resolvent for A, with 61
h b r Prj
pi H
Let the resolvent be
Bouf+..- + B,
(9)
and
let
0,
the expressions for the pt be
nm (10)
pt 59),
(see
=
=
Ea+'-'+EitS-iU?-
1 -
-
,
.p-'
where the Bi, Eg and
D
a
=
-,
1,
-,
r,
are simple forms in the
If the parametric (symbols for the) parametric derivatives. derivatives are specialized as functions of the Xi for which
B D^Q,
the functions pi determined by (9) and (10) give for the given (in some open region) all of the solutions of specialization of the parametric derivatives. Q
*
All
results
A
relative
to
simple
forms, which
without difficulty to several variables.
we
employ, carry over
ALGEBRAIC DIFFERENTIAL EQUATIONS
164
The
and (10) continue to hold
if the pi are the functions and (10) by appearing in (7). be given any vulues, close For, let the arguments in the to those in does not vanish. Let Q fy], for which
relations (9)
#
replaced in (8)
#
B D
,
If the parametric derivatives the value given to xt be JJ. are held fast at the values just assigned to them, while Xi then the pi in (7) beranges over the neighborhood of ,
come functions the
of
which, with the constant values
of the Xi,
parametric derivatives, give a solution of ^ 0, so that (9) and (10) hold.
this solution,
B D
A.
=
Now
For the
values of the pi in this solution, at Xi {, are the values of the functions gt in (7) with the arguments specialized as above. This shows that the gi can replace the pt in (8), (9),
(10).
We
thus consider each gi in (7) to be expressed by the second member of (10), where w is a function of the xi and
when
the parametric derivatives, analytic close to their values in It
may
be,
the arguments are
[*/].
,
however, that the expressions (10) are meaningis may vanish [*/]; that
D
less for the particular values ?,
To take care of this point, and of a point which will arise later, we pass to values ?', fa'], close to fy], for which DBQ K, where K is the discriminant of After we have proved the passivity (9), does not vanish.
for those values.
,
of (2) for the
neighborhood of
neighborhood of
,
Let equations be adjoined form an orthonomic system,
to
complete sets of monomials. a can be written. From (10) dpi
for the
*
(7),
as in
109, so as to
0, whose first members give Let us see how the equations
in
(11)
[VL the passivity
5',
fyl will follow.
we
find
_
dxj
(HiQ + Let
P
Then
+ g^-i^- + (Jo _+
represent the
1
)
first
member
+ e/M
of (9).
Let
Q
= dP/dw.
PARTIAL DIFFERENTIAL EQUATIONS
X.
(12)
|^
Now
=
the resultant of
g.
P
+ ... +
and
This means that
BK Q
Q
01^ B
equals, to within sign,
Q
K.*
= LP+MQ
and that the denominator Q in (12) can be replaced by B (we multiply the numerator by M). In the new expression for dw/dxj, the degree of the numerator in w may exceed s 1. We substitute this new expression for
K
Thus we have
into (11).
(13)
T
where the Fi are of
D,
We
B
,
free of
T
w, and
is
a product of powers
K. member
get expressions similar to the second
of (13)
for all derivatives of the pi. If principal derivatives appear At the end, in the Fi, we get rid of them, step by step.
we depress the degrees in w of the numerators in the expressions to less than s. This is accomplished by a division by P; the division introduces a power of Q into the deno-
B
minator. All in
all,
each equation in
dy
(14)
where
T
is
=
will
-^
have the form
T
a product of powers of
'
D,
B
,
K, where
are simple forms in the parametric derivatives. If we refer now to 118, we see that every
the neighborhood of
member
of (14).
To
',
[*/],
identically zero. *
Perron, Algebra, vol.
1,
has for
the passivity of (7), for the to show that every ^, as of the parametric derivatives, is
we have
neighborhood a function of the xi and
F
an expression like the second
establish
of ?, fa],
/*
the
p. 225.
ALGEBRAIC DIFFERENTIAL EQUATIONS
166
The form DB<>K, which involves only parametric is
reduced with respect to
and hence
(6)
derivatives,
not in 2.
is
Con-
DB
K^Q. any regular solution of (6) for which and the separants Let ?m be a point at which Q and initials do not vanish, for this solution. Let [*/"] represent sider
i'
,
DB K
,
the set of values, at ", ?m, for this solution, of the derivatives in (6). Let us imagine that we have formed the system (7) for the neighborhood of ?", IY']. Because the -
,
of the expressions for the fft, in (7), in terms involves w, only rational operations, the expressions will be the same for ?", [i?"] as for ?', [*/]. The same is true
calculation of
of the expressions for the p.
of
Suppose that the expression for some //>, say^, in terms w is not identically zero. Let Z be the numerator in
the expression for of (6).
^
P
Because
with respect to
.
Z vanishes for the
Then
w
W P W vanishes
W
for all solutions like the above,
the fact that
W involves
Thus the expression
is in
2.
This contradicts
only parametric derivatives.
^
for
is
Then
identically zero.
^i,
an analytic function, vanishes for the neighborhood of
as ?',
above solution and Z of
irreducible, the resultant As is not identically zero.
is
As
fa'].*
',
fy']
is arbitrarily
for the neighborhood of
,
close to
,
fy],
^
vanishes
[^].
This proves the passivity of (7). 126. Let (6), with AI not free of the unknowns, be an
ascending ditions
We
set.
to
for (6)
shall find necessary and sufficient conbe a basic set for a closed irreducible
system.
As a that
first
necessary condition, we have the condition as a set of simple forms, be a basic
when regarded
(6),
set for a
prime system. This implies the existence of r analytic functions gi, as in (7), which annul the Ai, when substituted for the pi, without annulling any initial or separant ( Let fy] be some set of values, as in ,
*
Note that not
all
numbers
in
f
'
[y
]
45).
125,
for
are arguments of the
/u
.
which
X.
no
initial
or
PARTIAL DIFFERENTIAL EQUATIONS
A
separant vanishes.
167
second necessary con-
dition is that the system (7) be passive for the neighborhood of Mil-
We forms,
shall prove that if (6), is
a basic
for a single
set
set ,
considered as a set of simple
of a prime
fy],
then (6)
system, is
and if
a basic
set
(7) is passive
of a
closed
ir-
reducible system.
Since (7)
is passive for the neighborhood of fy], the p 125 must vanish as analytic functions, for the neighborhood of ?, [97]. Hence the expressions of the p in terms of w, which are valid for the neighborhood of ?', [?'], vanish identi,
of
cally.
We
conclude that for any set of values
annul the AI but no
?,
[17]
at all
which
or separant, (7) is passive. The passivity of (7) for J, [ij\ as above implies that (6) has regular solutions. shall prove that the system 2 of initial
We
forms which vanish for
all regular solutions of (6) is an reducible system of which (6) is a basic set. Let O and be such that is in -5". Let G and
H
OH
be, respectively, the remainders of
O
and
H with
ir-
H^
respect to There may be, in Oi and (6). lf parametric derivatives not present in (6). But (6), considered as a set of simple
H
forms, will be the basic set of a prime system, 4, even after the adjunction of the new parametric derivatives to the un-
knowns
in the simple forms. passivity established above,
65, and using the see that every solution of which annuls no separant or initial in (6), leads, when considered at a quite arbitrary point of 21, to a regular solution
Following
A
we
of (6). Thus GiHi, considered as a simple form, is in A* Then one of G1} H^ is in A. As in 65, it follows that one of Gi,Hi vanishes identically. Then one of G, is in 2. Thus 2 is irreducible. What precedes shows that if
H
G
is in
Then
3, the remainder of a basic set of 3.
G
with respect to
(6) is zero.
(6) is
127. Given a set (6) which satisfies the
first
condition of
126, we can determine with a finite number of differentiations, rational operations and factorizations, whether or not (7) is 12
ALGEBRAIC DIFFERENTIAL EQUATIONS
168
This follows from the fact that the expressions of w can be formed by a finite number of
passive.
the
in terms of
/i
such operations. If (7) is not passive, the form
WDB
Q
K,
(as in
125),
which involves only parametric derivatives, vanishes for any regular solutions which (6) may have.*
ALGORITHM FOR DECOMPOSITION 128. Let
As
in
2
67,
be any finite system of forms, not
we can
by a
get,
finite
number
all
zero.
of differentiations,
rational operations and factorizations, a set, equivalent to 2, of
finite
systems,
-5*1,
,
2S
,
which have the following
properties: (a) The basic sets of each 2i are not higher than those of (b) if the basic sets of 2i involve
any form of 2i with respect to a basic set
of (c)
-2;
unknowns, the remainder is
zero
;
a basic set of 2i, considered as a set of simple forms, is a basic set of a prime system.
Suppose that 2t has a basic set (6), with A not free of unknowns. If (7) is not passive, 2^ is equivalent to
where
and Ii are the separant and initial of A*. Now the latter systems have basic sets lower than (6).
Si
of
all
If (7)
proves passive, 2^
where a basic It
is
number
is
is
equivalent to
the closed irreducible system of which (6)
is
set.
clear that
by
this
of steps, at a finite
process,
number
we
arrive,
in a
of ascending sets,
finite
which
are basic sets of a set of irreducible systems equivalent to 2. *It will be seen
which annul no
in
initial.
129 that
WDB K Q
vanishes for
all
solutions
X.
The above
PARTIAL DIFFERENTIAL EQUATIONS
169
a complete elimination theory for
constitutes
systems of algebraic partial differential equations. The test for a form to hold a system is as in
One
will notice that
68.
every system of linear partial differential
is irreducible.
equations
ANALOGUE- OF THE HILBERT-NETTO THEOREM 129.
We
theorem of
shall extend the
77 to the case of
several independent variables. As in the case of one variable, it suffices to show that if the system
Jl, Ft, ..-, Ft
(15)
has no solutions, then unity
and
of a certain
We
number
is
a linear combination of the F{,
of their partial derivatives.
One proves,
suppose that unity has no such expression.
as in the case of one variable, that there is a point at
which the
n power (16)
which certain
series
Cot
+ Cn (Xi
render each
We
coefficients in (15) are analytic, for
%,, amj
shall
Fj
use
Oi)
+
zero
when
this
fact
+ Cm
i
(Xm
substituted formally for y v yn . has that (15) analytic prove ,
to
solutions.
Let (15) be resolved into irreducible systems, by the method of 128. Here, we are dealing with analytic solutions, and not with formal ones. If we can show that one of the irreducible systems has a basic set in which the first form involves unknowns,
we
shall
know
that (15) has analytic
solutions.
Let us examine the process of decomposing (15) into
ir-
reducible systems, following 128. First, it is apparent that (16) is a formal solution of one of the systems ^.* Let (16) be a solution of li. Let (6) be a basic set of 2l .
2
In *The coefficients in the a*. may not be analytic at ai, that case, the coefficients are to be expressed as quotients of power series for ai Om. This will be possible, since the coefficients are meromorphic. t
,
,
,
ALGEBRAIC DIFFERENTIAL EQUATIONS
170
A iy
Then
passive, is
involves unknowns.
in (6),
If
then (15) has analytic solutions.
We
not passive.
the system (7) is Suppose that (7)
shall prove that
li--Ir
(17)
WDB K Q
Let us suppose that
vanishes for (16).
I,
.--Ir
does not vanish for (16). The system of simple forms
,
obtained by adjoining the
simple form (see (8))
to
A
of
126,
is
indecomposable. We are dealing here with The forms
analytic solutions of &.
hold &.
all
L vanishes
Thus,
if
L
is
any one
of the r
+1
forms (18),
for jevery analytic solution of the system of simple
forms AI,
(19)
,
Ar W ,
bipi
----- b r p r
Ir does not vanish. By the Hilbert-Netto which /i theorem for simple forms, some power of
for
is
a linear combination of the forms in
(19).
This means
L
vanishes for any formal power series solution of (19) lr does not vanish. for which 1^ that
Now,
let the pi
formally and let
w
satisfies (9)
If
of
w
be series obtained by differentiating (16) be the series given by
and that the
pi are given
(8).
by
We
see that
(10).
we go formally through the process of obtaining 126, we find that the expression for every ^ in
the
(*
terms
X.
of
w
PARTIAL DIFFERENTIAL EQUATIONS
vanishes for (16).
171
Then, if /MI, for instance, vanish for (16). Thus, if 2^ does not have a passive system (7), a solution of one of the systems
identically zero,
is
not
W must
(1-6) is
we find that (16) is a solution of a basic set of Then the first form irreducible system 2' held by (15). of this basic set must involve unknowns, so that 2' has
Continuing,
some
analytic solutions.
This completes the proof of the analogue, for partial difIt follows, forms, of the Hilbert-Netto theorem.
ferential
as in the case of one independent variable, that any
system
of
forms can be decomposed
into
finite
finite
irreducible
systems by differentiating the forms of tjie system a sufficient number of times and resolving the extended system, considered as a system of simple forms, into indecomposable systems.
INDEX The numbers
Adjunction of unknowns arbitrary
unknowns
refer to sections
17
Open region
24
Parametric derivative
108 108
ascending set
3
principal derivative
Basic set
4
Rank
Class of forms
2
relative reducibility
complete set
102
remainder
Extended
102
resolvent
set
Field
1
fields of constants
form
2,
algebraically irreducible
.
.
18
resultant
120
Separant
19
quotients
38
normal
simple
41
regular
Hilbert-Netto theorem
77,
initial
integrability conditions
Jacobi-Weierstrass form
Kronecker's theorem
Leader Luroth's theorem
Manifold
mark
seminormal
129
34 5,
121
5,
123
16
29
42 34 5,
123 6
solution
general
Indeterminate
2,
simple
34
general
1
19 72 20, 23,
125 72
system
123
closed
118
complete
71
35 121
14
indecomposable
41
orthonomic
6
non-trivial
106
7
essential irreducible
irreducible
91
23
passive
12
108
23 119
monomial
99
prime
42
multiple
99
simply closed
41
multiplier
103
172
systems equivalent
13
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