First Order Algebraic Differential Equations A Differential Algebrai
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804 Michihiko Matsuda
First Order Algebraic Differential Equations A Differential Algebraic Approach
Springer-Verlag Berlin Heidelberg New York 1980
Author Michihiko Matsuda Department of Mathematics, Kyoto Sangyo University Kamigamo, Kyoto 603/Japan
AMS Subject Classifications (1980): 12 H 05
ISBN 3-540-09997-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09997-2 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Introduction
The produced
study of first order a l g e b r a i c d i f f e r e n t i a l e q u a t i o n s
f r u i t f u l r e s u l t s a r o u n d the end of the
c l a s s i f i c a t i o n of e q u a t i o n s c a r r i e d out s u c c e s s f u l l y . the c o m p l e x p l a n e continuation". use of
free of m o v a b l e
last century.
s i n g u l a r i t i e s was
The i n v e s t i g a t i o n s w e r e c a r r i e d out in
and the m a i n tool of i n v e s t i g a t i o n was
Fuchs
The
"analytic
t r i e d to c l a r i f y the a l g e b r a i c a s p e c t m a k i n g
" P u i s e u x series",
but his w o r k was not d e v e l o p e d
fully at
that time. The m o d e r n t h e o r y of d i f f e r e n t i a l a l g e b r a and a l g e b r a i c function
fields of one v a r i a b l e has e n a b l e d us to give an a b s t r a c t
treatment,
l e a v i n g the c o m p l e x plane.
R e c e n t l y the a u t h o r pre-
sented a differential-algebraic criterion
for a first o r d e r alge-
b r a i c d i f f e r e n t i a l e q u a t i o n to have no m o v a b l e
s i n g u l a r i t y sug-
g e s t e d by Fuchs'
F r o m this stand-
criterion
p o i n t we r e c o n s t r u c t e d Bouquet,
for this property.
some c l a s s i c a l t h e o r e m s due to Briot,
F u c h s and Poincare.
In this t r e a t m e n t the c o e f f i c i e n t
field is an a r b i t r a r y a l g e b r a i c a l l y - c l o s e d d i f f e r e n t i a l characteristic
0.
E. R. K o l c h i n
, u s i n g G a l o i s t h e o r y of d i f f e r e n t i a l
o b t a i n e d in 1953 a t h e o r e m c o n t a i n i n g a c r i t e r i o n
§12).
The a u t h o r w o u l d
fields,
for a first or-
der a l g e b r a i c d i f f e r e n t i a l e q u a t i o n to d e f i n e e l l i p t i c (cf.
field of
functions
like to note that his work was m o t i v -
ated by this e x c e l l e n t theorem. t h e o r y to the p r o b l e m of e x p l i c i t
M. R o s e n l i c h t a p p l i e d v a l
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