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8]m
OSMANIA UNIVERSITY LIBRARY Call No.
Author
5/7'5g//f &7 /[
.
J)
Accession No.
V>
*
7 2--
J.
Title
(!),U llLi'Lt \^ 7 (^ be returned on or before the date BooliMiouId This
y
last
marked below*
AMERICAN MATHEMATICAL SOCIETY
COLLOQUIUM PUBLICATIONS, VOLUME
XIV
DIFFERENTIAL EQUATIONS FROM THE
ALGEBRAIC STANDPOINT
BY
JOSEPH PELS RITT PROFESSOR OF MATHEMATICS
COLUMBIA UNIVERSITY
NEW YORK PUBLISHED BY THE
AMERICAN MATHEMATICAL SOCIETY 501
WEST
116TH STREET 1932
Photo-Lithoprint Reproduction
KDWARDS BROTHERS, Lithoprinters
INC.
ANN ARBOR, MICHIGAN 1947
INTRODUCTION
We
shall
be concerned, in this monograph, with systems of
differential equations, ordinary or partial,
in the
unknowns and
their derivatives.
which are algebraic
The
algebraic side
such systems seems to have remained, up theory to the present, in an undeveloped state. It has been customary, in dealing with systems of differential of the
of
equations, to assume canonical forms for the systems.
Such
forms are inadequate for the representation of general systems. It is true that methods have been proposed for the reduction systems to various canonical types. But the limitwhich go with the use of the implicit function theorem, the lack of methods for coping with the phenomena of degeneration which are ever likely to occur in elimination processes and the absence of a technique for preventing the of general
ations
entrance of extraneous solutions, are merely symptoms of the futility inherent in such methods of reduction.
theory of systems of algebraic equations, one Kronecker's Festwitnesses a more enlivening spectacle.
Now,
in the
of 1882 set upon a firm foundation the theory of algebraic elimination and the general theory of algebraic manifolds. The contributions of Mertens, Hilbert, KOnig,
schrift
Lasker, Macaulay, Henzelt,
Emmy
and others, have brought, to
Noether, van der
Waerden
this division of algebra, a
high In the notions of irreducible manifold, and polynomial ideal, there has been material for far reaching On the formal qualitative and combinatorial investigations.
degree of perfection.
side,
one has universally valid methods of elimination and
formulas for resultants.
To bring
to the theory of systems of differential equations
which are algebraic
in the
unknowns and iii
their derivatives, *
INTRODUCTION
IV
some
completeness enjoyed by the theory of systems
of the
of algebraic equations, is the
The
aim
of the present
view which we take
of
point
that
is
of
monograph. our paper
Manifolds of functions defined by systems of algebraic differential equations, published in volume 32 of the Transactions
we
In what follows,
American Mathematical Society.
the
of
outline our results.
shall
Chapters I- VIII treat ordinary differential equations. We deal with any finite or infinite system of algebraic differential equations in the unknown functions y ly write each equation in the form
,
yn
of the variable
x
.
We
F(x\ y l9
F
-
-,y n )
=
0,
a polynomial in the yi and any number of their will be supposed to be derivatives. The coefficients in
where
is
F
An a given open region. All forms expression like F, above, will be called a form. considered in this introduction will be understood to have functions
of
coefficients
we mean
meromorphic
x,
in
which are contained
in a given field.
a set of functions which
a
field,
closed with respect to
and differentiation.*
rational operations
Let ~ be any
is
By
finite or infinite
system of forms in y^,
-
-,
yn
>
By a solution of 2, we mean a solution of the system of equations obtained by setting the forms of 2 equal to zero. The totality of solutions of 2 will be called the manifold of
2.
of
2
A
and
If ^\ is
l
22
a solution of
are systems such that every solution 2, we shall say that 22 holds 2 l9
system - will be called reducible or irreducible according and H, such that two forms,
as there do or do not exist
of 2,
G
GH
H
holds 2, while holds 2. The manifold and also the system of equations obtained by equating
neither
nor
the forms of
2
to zero, will be called reducible or irreducible
according as 2 is reducible or irreducible. We can now state the principal result of Chapter I. Every manifold is composed of a finite number of irreducible manifolds.
*A
formal definition
is
given in
1.
INTRODUCTION
V
given any system 2, there exist a finite number of 2S such that 2 holds every 2,, irreducible systems, 2 l9 -,
That
is,
-
,
while every solution of
is
-5"
The
a solution of some 2,.
decomposition into irreducible manifolds is essentially unique. Let us consider an example. The equation
-
()'-*'
>
=
2
whose
solutions are y a) (x (a constant), and y For is a reducible system in the field of all constants. ,
(9} \ A)
~i
ax
1
\
9
7 dx~
0,
9 &
first member of (1), while neither factor in (2) The system (1) is equivalent to the two irreducible
holds the does.
systems
dxl
=
-4yJ
0,
-
dx
=
and
The decomposition theorem follows from a lemma which bears a certain analogy to Hilbert's theorem on the existence
We
for an infinite system of polynomials. 2 that is an infinite system of forms in y\ prove y nj if then 2 contains a finite subsystem whose manifold is identical with that of 2.* of a finite basis
,
in
,
Chapters II and VI study irreducible manifolds. We start, Chapter II, with a precise formulation of the notion of
We
do not think general solution of a differential equation. Let A that such a formulation has been attempted before. be a form in y iy ducible, *
See
in
the
-,
given
yn
,
effectively involving
field,
as
yn
,
and
irre-
a polynomial in the yi and
124 for a comparison, with a theorem of Tresse, of the ex-
tension of this
lemma
to partial differential equations.
INTRODUCTION
yi
Let the order of the highest derivative let y w represent that derivative. Let 2 be the totality of forms which vanish for all solutions of A with dA/dynr ^O. We prove that 2 is irreducible. The manifold of 2 is one of the irreducible manifolds in the
their
derivatives.
of
in
yn
A
be r and
A
decomposition of the manifold of the general solution of
The remainder
of
A
(or of
A
.
=
We
call this
manifold
0).
Chapter II deals with the association,
with every irreducible system -2", of a differential equation which we call a resolvent of 2. The first member of the resolvent
is
an irreducible polynomial, so that the resolvent
has a general solution.
Roughly speaking, the determination
of the general solution of the resolvent
determination of the manifold of 2.
is
equivalent to the
The theory
of resol-
vents furnishes a theoretical method for the construction of all irreducible
One
systems.
be used advantageously
will see that the resolvent
can
formal problems. In Chapter VI, we study what might be called the texture of an irreducible manifold. For the case of the general solution of an algebraically irreducible form, our work amounts
to 9
those
characterizing
A 1 3 y nr
= 0)
Chapters
V
in
singular
solutions
algorithms, involving differentiations for
(solutions
which belong to the general solution. and VII contain, among other results,
decomposing a
finite
system into
with
finite
and rational operations, irreducible systems and
In Chapter V, we do not obtain the actual irreducible systems, but rather certain basic sets
for constructing resolvents. of forms (Ch. II)
which characterize the irreducible systems.
In However, permits the construction of resolvents. a is obtained if carried Chapter VII, sufficiently process which, this
far, will actually produce
the irreducible systems. Unfortunately,
nothing in this process which informs one, at any whether or not the process has had its desired effect. The results of Chapter V furnish a complete elimination
there
is
point, as to
theory for systems of algebraic differential equations. In Chapter VII, we derive an analogue, for differential forms, of the famous Nullstellensatz of Hilbert and Netto. In
INTRODUCTION
VU
Chapter VIII, we present an analogue of Liiroth's theorem on the parameterization of unicursal curves. In Chapter III, there will be found a theory of resultants of pairs of differ-
A
ential forms.
number
of other special results are distributed
through the monograph. In Chapter X, some of the main results stated above are extended to systems of algebraic partial differential equations. particular, an elimination theory is obtained for such
In
systems.
Chapter IV treats systems of algebraic equations. The chief purpose is to obtain special theorems, and finite algorithms,
The main equation theory. results of Chapter IV are known ones, but the treatment for
to
application
differential
appears new, and some special theorems, of importance for do not seem to exist in the literature.
us,
has been our aim to give this monograph an elementary character, and to assume only such facts of algebra and It
are
as
analysis
in
principle
of
exposition
contained in standard treatises.
With
we have devoted Chapter IX
mind,
Riquier's
remarkable
existence
this
to
orthonomic systems of partial differential equations. Thus Chapter IX is purely expository, and Chapter IV
The remaining chapters present
largely so.
an
theorem for is
results contained
above mentioned paper, and results communicated by us to the American Mathematical Society since the publication in our
of that paper.
Koenigsberger's irreducible differential equations,* and Drach's irreducible systems of partial differential equations,t In Drach's are irreducible in the sense described above. definition,
called
is
which includes that of Koenigsberger, a system if every equation which admits one
irreducible
the system admits all solutions of the system. Thus, systems which are irreducible in our sense may easily be reducible in the theories of Koenigsberger and Drach. solution
The *
of
definitions
of
Koenigsberger and Drach, which do not
Lehrbuch der Differenzialgleichungen, Leipzig, 1889.
t Annales de TEcole Normale,
vol. 34, (1898).
INTRODUCTION
Vlli
lead
to
into
decompositions
irreducible
systems,
are
the
starting points of group-theoretic investigations,
which parallel
we have
seen, is in a
the Galois theory.
Our
course,
as
different direction.
remain for investigation. In particular, differential forms and a theory of birational transformations, await development.* Chapters VII and VIII may perhaps be regarded as rudimentary beginnings
Many
a
of
questions
theory
of
still
ideals
of
such theories.
goes without saying that we have been guided, in our work, by the existing theory of algebraic manifolds. We have found particularly valuable, the excellent treatment of It
systems of algebraic equations given in Professor van der
Waerden's paper Zur Nullstellentheorie der PolynomidealeA But it is not surprising, on the other hand, that the investigation
new phenomena should have called for development of new methods. I am very grateful to the Colloquium Committee of of
essentially
the
the
American Mathematical Society, who have invited me to lecture on the subject of this monograph at the University of California in September, 1932. To my friend and colleague Dr. Eli Gourin, who assisted my deep thanks. *
me
in reading the proofs, I extend
In connection with transformations of general (non- algebraic) differMathematische Annalen, vol. 73 (1913), p. 95.
ential equations, see Hilbert,
t Mathematische Annalen,
NEW
vol. 96, (1927), p. 183.
YORK, N. Y.
February, 1932. J.
F. Rrrr.
CONTENTS Page
CHAPTER
I
DECOMPOSITION OF A SYSTEM OF ORDINARY ALGEBRAIC DIFFERENTIAL EQUATIONS INTO IRREDUCIBLE SYSTEMS .
Fields, forms, ascending sets, basic sets, reduction,
and manifolds, completeness of
infinite systems,
1
solutions
non-existence of
a Hilbert theorem, irreducible systems, the fundamental theorem, uniqueness of decomposition, examples, relative reducibility, adjunction of
new unknowns,
fields of constants.
CHAPTER
II
GENERAL SOLUTIONS AND RESOLVENTS
21
General solution of a differential equation, closed systems, arbitrary unknowns, the resolvent, invariance of the integer g, order of the resolvent, construction of irreducible systems, irreducibility and the open region 21.
CHAPTER
III
FIRST APPLICATIONS OF THE GENERAL THEORY
47
Resultants of differential forms, analogue of an algebraic theo-
rem
of
Kronecker, form quotients.
CHAPTER
IV
SYSTEMS OF ALGEBRAIC EQUATIONS
62
Indecomposable systems of simple forms, simple resolvents, basic sets of prime systems, construction of resolvents, resolution of a finite system into indecomposable systems, a special theorem.
CHAPTER V CONSTRUCTIVE METHODS
92
Characterization of basic sets of irreducible systems, basic sets in a resolution of a finite system into irreducible systems, test for a form to hold a finite system, construction of resolvents, a remark
on the fundamental theorem, Jacobi-Weierstrass canonical form. ix
CONTENTS
X
Page
CHAPTER
VI
CONSTITUTION OF AN IRREDUCIBLE MANIFOLD Seminorraal solutions, adjunction of composability and irreducibility.
CHAPTER
new
100
functions to o?, inde-
VII
ANALOGUE OF THE HILBERT-NETTO THEOREM.
THEORETICAL
DECOMPOSITION PROCESS
.
of Hilbert-Netto
theoretical
.
108
for
theorem; Analogue process decomposing a finite system of forms into irreducible systems; forms in one unknown, of first order.
CHAPTER ANALOGUE FOR FORM
VIII
QUOTIENTS OF MROTH'S THEOREM.
124
CHAPTER IX RIQUIER'S EXISTENCE
THEOREM FOR ORTHONOMIC SYSTEMS 135
Monomials, dissection of a Taylor
series,
marks, orthonomic
systems, passive orthonomic systems.
CHAPTER X SYSTEMS OF ALGEBRAIC PARTIAL DIFFERENTIAL EQUATIONS 157 Decomposition of a system into irreducible systems, basic sets of
closed
irreducible
systems,
algorithm
analogue of the Hilbert-Netto theorem.
for
decomposition,
CHAPTER
I
DECOMPOSITION OF A SYSTEM OF ORDINARY ALGEBRAIC DIFFERENTIAL EQUATIONS INTO IRREDUCIBLE SYSTEMS FIELDS
We
i.
consider functions meromorphic in a given open the plane of the complex variable x*
We
in
21
region
recall that
an open region
every point of the set
(a)
radius, all of
is
is
a set of points such that the center of a circle of positive
whose points belong
to the set;
any two points of the set can be joined by a continuous
(b)
curve whose points
A
set
r,
all lie
in the set.
of functions described as above,
a field if (a) & contains at least one function which
is
will
be called
not identically zero;
f
and g (distinct or equal), be7 longing to *& then fiLg and fg belong to given any two functions, f and g belonging to oF, ivith g not identically zero, then fig belongs to 7 given any function, f, in $r, the derivative of f belongs any two functions
(b) given
,
(c)
y
(d)
to
3.
Every
field
contains
all
rational constants.
Examples
of
the totality of rational constants; the totality of rational functions of &; all rational combinations of a: and e*
fields are:
with constant coefficients;
all
elliptic functions
with a given
period parallelogram.t *
We
are dealing here only with the finite plane. of analytic functions has appeared
fThe notion of field among other places, in
Picard's group-theoretic 1
previously, investigations on linear
ALGEBRAIC DIFFERENTIAL EQUATIONS
2
FORMS In what follows,
2.
which
is
supposed
we work
with an arbitrary
field
cF,
be assigned in advance and to stay
to
fixed.
We
are going to develop some notions in preparation for study of differential equations in n unknown functions,
the Vi>
-->yn
-
By a differential form or, more briefly, by a form, we shall understand a polynomial in the y i and any number of their derivatives, with coefficients
With respect
to every
meromorphic in 21. form introduced into our work, we
shall assume, unless the contrary is stated, that its coefficients
belong to S
r .
Differentiation
yt
will be indicated
by means
Thus
second subscript.
of a
We
of functions
=y
write, frequently, y.
io
.*
Throughout cur work, capital italic letters will denote forms. By the jth derivative of A, we mean the form obtained by differentiating
Af
times with respect to x, regarding y t
,
,
yn
as functions of x.
the order of A with respect to yi, if A involves yi or some of its derivatives effectively, we shall mean the greatest,;'
By
differential equations
and in Landau's work on the factorization of linear
differential operators.
The foregoing
p. 562.
See Picard, writers
Traite d'Analyse, 2nd edition, vol. the additional assumption that
make
3, r
Loewy, however, in his work on systems of linear differential equations, Mathematische Aiinalen, vol. 62 (1906), p. 89, does not make this additional assumption. No generality would be gained by contains
allowing
With
all
constants.
to consist of functions analytic except for isolated singularities.
this assumption,
it
is
an easy consequence of Picard's theorem on
essential singularities, and of the fact that
contains all rational constants, that the functions in c7 are meromorphic. * In certain problems, we shall use unsubscripted letters to represent unknowns. If y is such an unknown, will represent the jth derivative y^
of y.
r
IRREDUCIBLE SYSTEMS
I.
3
present in a term of A with a coefficient If A does not involve j/$, the order of A distinct from zero.
such that
is
yij
with respect to
be taken as 0.
yi will
By the class of A, it A involves one or more yi effectively, we shall mean the greatest p such that some ypj is effectively present in A.
If
A
A
simply a function of x,
is
will
be said
to be of class 0.
Let AI and A* be two forms. If A 2 is of higher order than Ai in some y p A2 will be said to be of higher rank If A and A 2 than A l9 and A^ of lower rank than A 2 in y p are of the same order, say q, in yp and if A 2 is of greater ,
.
,
A
degree than
l
in
higher rank than
of
difference
in
rank
*
ypq
A
is
then, l
in
again,
yp
A2
Two
.
established
will be said to be
forms for which no
by the foregoing
criteria
be of the same rank in yp If A 2 is of higher class than A i9 A 2 will be said to be of higher rank than A\A If A 2 and AI are of same class p>0, and if A 2 is of higher rank than A t in yp then, again, A 2 will will be said to
.
,
be said to be of higher rank than AI. Two forms for which no difference in rank is created by the preceding, will be said to be of the
A2
same rank.|
As
higher than AI, A it than higher If
is
higher than
A 2j
then A$
is
we shall have occasion to use other yn for the unknowns. If the unknowns are given in the order u,v, -, w, then, in the definitions of class and of relative rank, the ^th unknown from the left is to be treated like y p above. We shall need the following lemma: LEMMA. If In later chapters,
symbols than yl9
-,
AI
*
,
A
2
,
,
Aq,
If a form is identically zero (hence Considered as a polynomial in ypq y^ it will be considered of degree in every ypo. .
of order zero in every
This leads to no
fWe
difficulties.
"At is higher than forms of class zero are of the same rank.
shall frequently say, simply,
JThus,
all
A
v
".
ALGEBRAIC DIFFERENTIAL EQUATIONS
4
an
is
Aq
such that, for every q, Aq+i a subscript r, such that, for
infinite sequence
Aq
higher than
,
there exists
Ar
has the same rank as
The
Aq
classes of the
not
is
q>r,
.
form a non-increasing set of non-
It is then clear that, for q large, the negative integers. A q have the same class, say p. If p >0, the A q with q large
be of the same order, say
will
An
s, in
common degree
eventually have a
immediate
consequence
finite or infinite aggregate of
not higher than any other
Finally, the Aq yp yps
of
will
.
in
.
this
lemma
forms contains a
is
that every
form
ivhich is
the aggregate.
form of
ASCENDING SETS 3. If
AI
of class
is
with respect
A
to
if
^)>0,
A2
will
^4.2
be said to be reduced
lower rank than AI in yp
of
is
.
The system
A A2
(1)
t ,
be called an ascending
will (a)
r =
1
>
,
,
Ar
set if either
and A ^ l
or (b) r
>1,
AI
is
of class greater than 0, and, for
class
than Ai and reduced
of higher Of course, r i,
respect
Aj
is
to Ai.
said to be of higher rank
set (1) will be
than the ascending set J3 lf
(2) if
,,
ft,
either
There
(a)
is
a j. exceeding neither r nor
Bi are of than BJ*
the
same rank for
i
<j
s,
and
such that Ai and that Aj is higher
or
same rank for i < r. Two ascending sets for which no difference in rank is created by what precedes will be said to be of the same
s>r
(b)
*
If
j
=
and Ai and Bi are of
1
,
this is to
mean
that AI
the
is
higher than BI
.
I.
For such
rank.
rank for every Let !, 2
We
,
tf> 3
2
let
tf> 2
<J>i
>
is
i
O^^,
.
and
0>i
and At and Bi are of the same
s
i.
shall prove that
Let
and
2
O
=
sets, r
5
be ascending sets such that We write higher than 8
be represented by
and
(1)
(2) respectively
be
8
'
*
Cl>
/2
'
Lt
7
>
>
for the reason (a) and that 2 Suppose first that i Let reason for the #8 (a). 2 j be the smallest integer Then either Ai is of the than is such that Bj higher Q-. or there is a k
2 >0>8 by (a). Suppose It j r, i 8 by (b).
>
i> 8
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
*
3
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 g for any q. such that, for q>r, O q has the same rank as r fe
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 pjpj.
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 ?i + 0>i) + *i and 2 + F2 holds Now every solution of (0>! + i + 0> 8 ) + F2 .*
Let 2 + Ft
<J>
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 , the system 2 of all forms which hold is closed, and has the same manifold as 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
^'
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