Galois action on solutions of a differential equation
type: Item
instance of: scholarly article
title: Galois action on solutions of a differential equation
zbMATH Open document ID: 0855.12006
author: Peter A. Hendriks
author: M. van der Put
published in: Journal of Symbolic Computation
publication date: 1996-12-03T00:00:00Z
full work available at URL: 1995JSymbCompHendriks.pdf
review text: To a linear differential equation \(L\) with coefficients in \(\mathbb Q (x)\), one can associate a Riccati equation \(R\) whose solutions are the logarithmic derivatives of solutions of \(L\). The aim of this paper is, under the hypothesis that \(R\) has a rational (resp. algebraic) solution, to give a bound for the degree of an extension \(C\) of \(\mathbb Q\) such that \(R\) has a solution in \(C(x)\) (resp. has an algebraic solution whose coefficients of the monic irreducible polynomial are in \(C(x)\)). The answer is complete for second-order equations and almost complete for third-order equations (in that case no example showing the bounds to be sharp is available). The first part of the paper is devoted to stating basic facts in differential Galois theory over the differential field \(k(x)\) or \(k((x))\) when \(k\) is a non necessarily algebraically closed field of characteristic 0. Bounds for the degree of \(C\) are given in the second part. They rely on a classification due to \textit{J. Kovačić} for second-order equations [J. Symb. Comput. 2, 3--43 (1986; Zbl 0603.68035)] and to Singer and Ulmer for third-order equations. Examples showing that bounds are sharp are given in the third part.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34-04
zbMATH DE Number: 845990
zbMATH Keywords: linear differential equation
zbMATH Keywords: Riccati equation
zbMATH Keywords: bound for the degree of an extension
zbMATH Keywords: differential Galois theory
zbMATH Keywords: differential field
zbMATH Keywords: second-order equations
zbMATH Keywords: third-order equations
Wikidata QID: Q115395348
reviewed by: Gilles Christol
MaRDI profile type: Publication
OpenAlex ID: W1969267425
DOI: 10.1006/JSCO.1995.1032
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description: scientific article; zbMATH DE number 845990