Shidlovskii irreducibility
type: Item
instance of: scholarly article
title: Shidlovskii irreducibility
zbMATH Open document ID: 0827.12006
DOI: 10.1016/0019-3577(94)90017-5
author: Peter A. Hendriks
published in: Indagationes Mathematicae. New Series
publication date: 1995-12-17T00:00:00Z
review text: Consider an \(n\times n\)-system of linear differential equations \(dY/dz= AY\) with \(A\in M_n (\mathbb{C} (z))\) and \(Y= (y_1, \dots, y_n)^t\) the vector of unknown locally complex analytic functions. In proving transcendence of the values of solutions at algebraic points Shidlovskij introduced the concept of (Shidlovskij) irreducibility. The system is said to be linearly Shidlovskij irreducible if the non-zero components of any solution vector are \(\mathbb{C}(z)\)-linear independent. In this paper the author gives a criterion for such irreducibility in terms of the structure of the differential Galois group of the system. This extends earlier results by Beukers, Brownawell, Heckman on the characterisation of Siegel normality.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 749792
zbMATH Keywords: Shidlovskij irreducibility
zbMATH Keywords: system of linear differential equations
zbMATH Keywords: transcendence
zbMATH Keywords: values of solutions at algebraic points
zbMATH Keywords: differential Galois group
reviewed by: Frits Beukers
MaRDI profile type: Publication
full work available at URL: 0019-3577(94)90017-5
OpenAlex ID: W4206726904
cites work: Siegel normality
cites work: Q3358852
cites work: Q3255437
cites work: Algebraic Groups and Algebraic Dependence
cites work: Q5680222
cites work: Differential Galois theory and tensor products
cites work: Transcendental numbers. With a foreword by W. Dale Brownawell. Transl. from the Russian by Neal Koblitz
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description: scientific article; zbMATH DE number 749792