First order algebraic differential equations: A differential algebraic approach (Lecture notes in mathematics)
by: Michihiko Matsuda
Book Preface (Contents outlined):
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 presented a differential-algebraic criterion for a first order algebraic differential equation to have no movable singularity suggested by Fuchs1 criterion for this property. From this standpoint we reconstructed some classical theorems due to Briot, Bouquet, Fuchs and Poincar^. 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 order algebraic differential equation to define elliptic functions (cf. 112). The author would like to note that his work was motivated by this excellent theorem. M. Rosenlicht applied valuation theory to the problem of explicit solvability of certain algebraic 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 introduced: 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
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