Fuchs inequalities for systems of linear differential equations with regular singular points
type: Item
instance of: scholarly article
title: Fuchs inequalities for systems of linear differential equations with regular singular points
zbMATH Open document ID: 1157.34064
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-09-03T00:00:00Z
review text: Let \[ \frac{{dy}}{{dz}} = B(z)y\tag{1} \] be a system of \(p\) linear differential equations with the matrix \(B(z)\) whose entries are meromorphic on the Riemann sphere \(\bar C\) and holomorphic outside the set of singular points \(a_1 , \dots ,a_n.\) Suppose also that the matrix \(B(z)\) has the following Laurent expansion in a neighborhood of a singular point \(a_i \neq \infty\) : \[ B(z) = \frac{{B_{ - r_i - 1}^i }}{{(z - a_i )^{r_i + 1} }} + \cdots + \frac{{B_{ - 1}^i }}{{z - a_i }} + B_0^i + \cdots \] The author refines the Fuchs inequalities for system (1) obtained by \textit{E. Corel} [Inégalités de Fuchs pour les systèmes differéntiels réguliers, C. R. Acad. Sci. Paris, Sér. I,328, 983--986 (1999; Zbl 0983.34081)). Theorem. For the sum \(\Sigma\) of exponents of the system (1) with regular singular points \(a_1 , \ldots ,a_n \) of Poincaré rank \(r_1 , \dots ,r_n \), respectively there holds the inequality \[ -\frac{{p(p - 1)}}{2}\sum\limits_{i = 1}^n {r_i } + \sum\limits_{i = 1}^{n}{ \frac{{k_i (k_i - 1)}}{2}} \leq \Sigma \leq - \sum\limits_{i = 1}^n {\text{ rank}} B_{ - r_i - 1}^i r_i, \] where \(k_i = p - {\text{ rank}} B_{ - r_i - 1}^{i} = \dim \ker B_{ - r_i - 1}^{i} \) if \(r_i > 0\) and \(k_i = 0\) if \(r_i = 0\).
reviewed by: Mykola Grygorenko
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 5319787
zbMATH Keywords: Fuchs inequalities
zbMATH Keywords: regular singular points
author: R. Gontsov
MaRDI profile type: Publication
full work available at URL: s10958-007-0339-3
OpenAlex ID: W2043252669
cites work: Q4295769
cites work: Q4785247
cites work: Q5616341
cites work: Q5737197
DOI: 10.1007/S10958-007-0339-3
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description: scientific article; zbMATH DE number 5319787