Versal unfolding of irregular singularities of a linear differential equation on the Riemann sphere
type: Item
instance of: scholarly article
title: Versal unfolding of irregular singularities of a linear differential equation on the Riemann sphere
zbMATH Open document ID: 1498.34237
DOI: 10.4171/PRIMS/57-3-6
author: Toshio Oshima
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2021-12-15T00:00:00Z
review text: The author investigates the possibility to realize a linear differential operator \(P\) with unramified irregular singular points on the Riemann sphere as a confluence of singularities of a Fuchsian linear differential operator (called an unfolding of \(P\)) having the same index of rigidity. It is conjectured that this is always possible. There are given self-contained proofs of some known auxiliary results, such as the decomposition theorem for \(P\) or theorem describing basic formal solutions of the equation \(Pu=0\). Also, formal characteristic exponents of \(P\) at its singular point are defined and an analytic proof of the Fuchs-Hukuhara relation for their sum over all singular points is given, as an alternative to an algebraic approach of \textit{D. Bertrand} and \textit{G. Laumon} [``Appendice à exposants, vecteurs cycliques et majorations de multiplicités'', C. R. Conf. Franco-Japonaise, Preprint IRMA (1985)]. Then the definitions of generalized characteristic exponents (of generalized Riemann scheme) of \(P\) and of the spectral type of \(P\) are given. The latter determines the index of rigidity of \(P\). The conjecture is checked to be affirmative under some assumptions on the index of rigidity and spectral type of \(P\). Several examples of unfolding are presented.
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34M25
zbMATH DE Number: 7445180
zbMATH Keywords: irregular singularity
zbMATH Keywords: Fuchsian singularity
zbMATH Keywords: confluence of singularities
Wikidata QID: Q114021400
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7445180