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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)
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


Applied Mathematics, General Mathematics