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Publication . Article . 1976

Solutions of linear differential equations in function fields of one variable

Michael F. Singer;
Open Access
Published: 01 Jan 1976 Journal: Proceedings of the American Mathematical Society, volume 54, pages 69-72 (issn: 0002-9939, eissn: 1088-6826, Copyright policy )
Publisher: American Mathematical Society (AMS)
Abstract
Formal power series techniques are used to investigate the algebraic relationships between a function satisfying a linear differential equation and its derivatives. We are able to derive some conclusions, among them that an elliptic function satisfies no linear differential equation over a liouvillian extension of the complex numbers. In [3], Rosenlicht noticed that if an element y belonged to a liouvillian extension of a differential field, then the zeroes and poles of it and its derivatives must satisfy certain relations. His main tool was THEOREM. Let K be a field of characteristic zero, k a subfield of K, P a real discrete k-place of K whose residue field is algebraic over k, D a derivation of K that is continuous in the topology of P and that maps k into itself. Let x, y be nonzero elements of K such that each of x(P), y(P ) is either 0 or xo. Then: (1) If ordp(Dx/x) _ 0, then ordp(Dy/y) ' 0. Here D induces a derivation on the residue field of P. Denoting this residue field derivation by the same symbol D, for any z in K such that ordp z ' 0, we have (Dz) (P ) = D(z(P )). (2) If ordp(Dx/x) such that the derivation ' is continuous in the topology of this place. Then ordp w < 0 implies that ordp(w'lw) _ 0. Received by the editors January 9, 1974 and, in revised form, December 5, 1974. AMS (MOS) subject classifications (1970). Primary 12H05. ' American Mathematical Society 1976
Subjects by Vocabulary

Microsoft Academic Graph classification: Formal power series Universal differential equation Mathematical analysis Linear function (calculus) Field (mathematics) Linear differential equation Residue field Algebraic differential equation Mathematics Pure mathematics Elliptic function

Subjects

Applied Mathematics, General Mathematics

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