On super-irreducible forms of linear differential systems with rational function coefficients.
type: Item
instance of: scholarly article
title: On super-irreducible forms of linear differential systems with rational function coefficients.
zbMATH Open document ID: 1070.34119
published in: Journal of Computational and Applied Mathematics
publication date: 2004-01-26T00:00:00Z
review text: Let \(F\) be a subfield of the field \(\mathbb C,\) and consider the system of equations \[ \frac{dy}{dx}=A(x)y, \] where \(A(x)\) is a matrix whose entries are in \(F(x)\) (the field of rational functions with coefficients in \(F\)). Suppose that \(x=\alpha\in \mathbb C\) is a pole of \(A(x).\) \textit{J. Moser} [Math. Z. 72, 379--398 (1960; Zbl 0117.04902)] gave an algorithm to decide whether \(x=\alpha\) is a regular singularity. This algorithm yields a coefficient matrix called the Moser irreducible form, which has the minimal pole order among matrices equivalent to \(A(x)\) under transformations \(y=T(x)z\) with entries in \(F(\alpha)[x].\) The notion of super-irreducible forms, which was introduced by \textit{A. Hilali} and \textit{A. Wazner} [Numer. Math. 50, 429--449 (1987; Zbl 0627.65091)], is a generalization of that of Moser irreducible forms. Let \(\Gamma=\{\alpha_1,\dots, \alpha_d\}\subset \mathbb C\) be a set of poles of \(A(x)\) which are the roots of some irreducible polynomial \(p(x)\) in \(F[x].\) In this paper, the author proposes an algorithm for computing an equivalent system with coefficients in \(F(x)\) which is super-irreducible in each \(\alpha_j\in\Gamma.\) In this algorithm, all computations are performed in \(F[x]/(p).\)
reviewed by: Shun Shimomura
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 2032609
zbMATH Keywords: linear differential systems
zbMATH Keywords: regular and irregular singularities
zbMATH Keywords: super-irreducible
zbMATH Keywords: forms
zbMATH Keywords: Moser-irreducible forms
author: Moulay A. Barkatou
MaRDI profile type: Publication
full work available at URL: j.cam.2003.08.020
OpenAlex ID: W1993448820
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DOI: 10.1016/J.CAM.2003.08.020
description: scientific article; zbMATH DE number 2032609