Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold dna .B Eckmann
804
Michihiko Matsuda
First Order Algebraic
Differential Equations
A Differential Algebraic Approach
galreV-regnirpS
Berlin Heidelberg New York 1980
Author
Michihiko Matsuda
Department of Mathematics, Kyoto Sangyo University
Kamigamo, Kyoto 603/Japan
AMS Subject Classifications (1980): 21 H 05
ISBN 3-540-09997-2 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-09997-2 Springer-Verlag NewYork Heidelberg Berlin
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Introduction
The study of first order algebraic differential equations
produced fruitful results around the end of the last century. The
classification of equations free of movable singularities was
carried out successfully. The investigations were carried out in
the complex plane and the main tool of investigation was "analytic
continuation". Fuchs tried to clarify the algebraic aspect making
use of "Puiseux series", but his work was not developed fully at
that time.
The modern theory of differential algebra and algebraic
function fields of one variable has enabled us to give an abstract
treatment, leaving the complex plane. Recently the author pre-
sented a differential-algebraic criterion for a first order alge-
braic differential equation to have no movable singularity sug-
gested by Fuchs' criterion for this property. From this stand-
point we reconstructed some classical theorems due to Briot,
Bouquet, Fuchs and Poincare. In this treatment the coefficient
field is an arbitrary algebraically-closed differential field of
characteristic .0
E. R. Kolchin , using Galois theory of differential fields,
obtained in 1953 a theorem containing a criterion for a first or-
der algebraic differential equation to define elliptic functions
(cf. §12). The author would like to note that his work was motiv-
ated by this excellent theorem. M. Rosenlicht applied valuation
theory to the problem of explicit solvability of certain algebraic
VI
differential equations successfully.
In this note we shall consider the general case in which
the coefficient field is an arbitrary differential field: It is
not necessarily of characteristic 0 nor algebraically closed.
We assume the reader to be familiar with the contents of the first
six chapters of the book "Introduction to the theory of algebraic
functions of one variable" by Chevalley (Amer. Math. Soc. 4th
printing, 1971), which will be referenced as [C]. Any theorem
not contained in this book and used here will be proved, even if
the proof is well known. A familiarity with differential algebra
is not assumed except in §18.
In §§16-17 recent results of Keiji Nishioka will be in-
troduced: They are valid only in the case of characteristic 0.
The author would like to express his sincere gratitude to
Professor I. Laine and Professor M. Rosenlicht for their invitation
to the Colloquium on Complex Analysis, Joensuu, Finland, August
24-27, 1978 and the Special Lecture at the University of California,
Berkeley, July 18, 1978 respectively.
August 1979
Michihiko Matsuda
Preliminaries
Let M be a commutative ring and N be a subring of M. A
derivation of N into M is a mapping 6 which satisfies the fol-
lowing conditions:
~(y + )z = ~y + ~z, ~(yz) = y~z + z~y.
A derivation of M into itself is called simply a derivation of M:
If M has no proper zero divisors, then it can be extended to
a derivation of the quotient field of M in one and only one way.
Suppose that M is a field and N is a subfield of M. The
derivation @ of N into M satisfies
6(y/z) = (z 6y - y6z)/z 2 z ~ 0
r •
If two derivations of N into M coincide on some subset E of N,
then they coincide on the subfield of N generated by the elements
of E.
Let x be an element of the field M. If x is transcenden-
tal over the subfield N, the derivation 6 can be extended to a
derivation D of N(x) into M such that Dx is an arbitrarily chosen
element of M(cf. §i, p.~). Suppose that x is algebraic over N.
If x is separable over N, ~ can be extended to a derivation of
N(x) into M in one and only one way. If an element u of N is the p-th
power of x, 6 can be extended to a derivation D of N(x) into M if
and only if 6u = 0, where p is the characteristic of N. In this
case an arbitrary element of M may be taken for Dx if x ~ N(cf.
for instance pp.12-13 of the book "Foundations of algebraic geom-
etry" by Weil(Amer. Math. Soc., 2nd Edition, 1962)).
Table of Contents
§i. Differential algebraic function field of one variable .... 1
§2. Movable singularities .................................... 7
§3. Fuchs' theorem ........................................... ii
§4. Extensions of the coefficient field ...................... 17
§5 Briot-Bouquet's theorem .................................. 22
§6 Clairaut fields .......................................... 27
§7 Poincar4 fields .......................................... 33
~8 Analogs of addition formulae for elliptic functions ...... 42
§9 Poincar4 fields generated by the constants ............... 47
§i0 Poincar4 fields of characteristic 0 ..................... 52
§ii. Automorphisms of an elliptic field ...................... 56
§12 Differential automorphisms .............................. 69
§13 Differential automorphisms of a Poincar4 field .......... 74
§14. Riccati fields of characteristic 0 ...................... 78
§15. Weierstrass points ...................................... 83
§16. Clairaut fields of characteristic 0 ..................... 91
§17. Movable singularities in case of characteristic 0 ....... 95
§18. Differential Lefschetz principle ........................ 103
Bibliography .................................................. ll0
Index ......................................................... iii
§i. Differential algebraic function field of one variable
A differential field is a field to which a derivation of
it is attached. The affixed derivation is called the differen-
tiation and signed with the prime "'"
An element of a differential field whose derivative van-
ishes is a constant. The totality of constants forms a field, the
field of constants. If the characteristic p of a differential
field is positive, the p-th power of any element is a constant.
Hence, a perfect differential field consists of constants if p is
positive.
A subfield of a differential field is called a differen-
tial subfield if it is closed under the differentiation. The
overfield is called a differential extension of it.
An algebraic function field of one variable over a field
K is a field R containing K as a subfield which satisfies the
following condition: There is an element x of R which is trans-
cendental over K, and R is algebraic of finite degree over K(x).
The coefficient field of R is the totality of those elements of
R which are algebraic over K: It is the one called the field of
constants of R in [C]: This term is used here for the meaning
stated ahead.
In case R is of the form K(x, y), the coefficient field
of R coincides with K if the characteristic polynomial f(Y) of
y with respect to K(x) is irreducible in K(x) [Y], where K is the
algebraic closure of K: For, if u is an element of R which is
algebraic over K, we have
[R : K(x)] = [R : ~(x, u)],
whence u E K(x): Since K is relatively algebraically closed in K(x),
u belongs to K. In case R is separably generated over K, the
converse is true; the coefficient field K* of R contains K proper-
ly if f(Y) is reducible in K(x)[Y] : For there are an element u of
and an intermediary field L between K and K such that f(Y) is
irreducible in L(x)[Y] but reducible in L(u, x)[Y]: Let g(U) be
the characteristic polynomial of u in L(u) with respect to L: It is irre-
ducible in L(x)[Y] because L is relatively algebraically closed
in L )x( , whence
[L(u, x) : L(x) ] = [L(u) : L] :
We set T = R(L) and consider it an algebraic function field of one
variable over L: The coefficient field of T is L if L contains K*,
since R is assumed to be separably generated over K([C, p.91]):
If K* were K, L would contain K*, whence L would be relatively
algebraically closed in T and g(U) would be irreducible in T[U]:
We would have
[T(u) : T] = [L(u) : L] ,
but it is impossible because
[T(u) : L(u, x)] < [T : L(x)]
by our assumption that f(Y) is reducible in L(u, x)[Y]. In the
general case our converse is not true; we can find a counter ex-
ample in [C, p.91]
An algebraic function field R of one variable over K will
be called a differential algebraic function field of one variable
over K if R is a differential field in which K is closed under
the differentiation.
In case R is separably generated over K, the coefficient
field of R is closed under the differentiation, because it is
algebraic and separable over K. In the general case it is not
true: Assume for instance that R = K(a, x), where K is a field of
positive characteristic p and x is transcendental over K, while
p ~ a K, a ~ K. If we define
x' = 0, a' = x, c' = 0, c 6 K,
R is a differential algebraic function field of one variable over
K, but the coefficient field K(a) of R is not closed under the
differentiation.
Let R be an algebraic function field of one variable over
a given field K. We shall suppose that K is a differential field
and see how R becomes a differential extension of K under the as-
sumption that R is separably generated over K. For this purpose
take a separating variable x in R: We shall show that for any e-
lement y of R there is a derivation D of K(x) into R such that
Dx = y and it coincides with the given derivation on K. If ~ is
i
an element of K[x] of the form Za.x , we define
1
!x i + y~ia.x i-1
D~ = ~a I 1
Clearly it is a derivation of K[x] into R, which satisfies
Dx = y, Da = a', a ~ K.
We may extend it to a derivation D of K(x) into R by
D(w) = (~D~ - #D~)~ -2, w = ~/~, ~, ~ ~ K[x];
it is well-defined on K(x). The derivation D thus defined can be
extended to a derivation of R in one and only way, since R is
algebraic and separable over K(x).
This being said, let f(X, Y) be an irreducible polynomial
with the coefficients in a given field K such that fy(X, Y) # 0.
If K is a differential field, there is a differential algebraic
function field R of one variable over K such that R = K(x, x')
with f(x, x') = 0. In case f(X, Y) is irreducible in K[X, Y],
the coefficient field of R is K.
Let R be a differential algebraic function field of one
variable over K. We shall show that if ~ is a place in R, the
differentiation is continuous on R in the topology induced by ~ .
In case the coefficient field K* of R is closed under the dif-
ferentiation, the proof is found in [C, p.l14]. In the general
case the proof goes through if the number o s presented there is
replaced by O + s Sl, where I s is the number defined below. Let
{el, ... , ~r } be a base of K* with respect to K. We set
I = s max {0, -~ ~(~) ; 1 ~ i ~ r}:
here ~ is the order function at ~. If a is an element of K*;
a = Zc.~. (i < i < r), c. 6 K,
l l = = 1
then we have
a' = Zci~ i + Zci~,
whence
~(a') ~ min {~ ~(~); 1 ~ i ~ r} ~ 1 - s
because c! 6 K.
1
This being said, if ~ is a place in a differential alge-
