On \(d\)-solvability for linear differential equations
type: Item
instance of: scholarly article
title: On \(d\)-solvability for linear differential equations
zbMATH Open document ID: 1166.12004
author: Nguyen An Khuong
published in: Journal of Symbolic Computation
publication date: 2009-04-14T00:00:00Z
review text: Let \(F\) be an ordinary differential field of characteristic zero, \(L(y)=0\) be a linear differential equation of degree \(n\) and \(d\) be a natural number less then \(n\). In the paper the following question, originally posed by \textit{L. Fuchs} [Act. Math. 1, 321--362 (1883; JFM 15.0256.01)] is discussed. Is it possible to express all solutions of the equation \(L(y)=0\) in terms of solutions of homogeneous linear differential equations of order less than or equal to \(d\)? The author specifies the statement of the problem within the frames of differential Galois theory. For this purpose he introduces the concept of \(d\)-solvability for Picard-Vessiot extensions and linear algebraic groups. Then he proves the equivalence of these concepts within differential Galois theory. The main result of the paper is a theorem connecting the \(d\)-solvability of a connected semisimple algebraic group with the minimal dimensions of representations of simple subalgebras of its Lie algebra. The results of the paper can be useful, for example, in number theory in the proof of algebraic independence of values of \(E\)-functions [see the reviewer, Mat. Zametki 34, No. 4, 481--484 (1983; Zbl 0548.12014)].
reviewed by: Mykola Grygorenko
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 5543812
zbMATH Keywords: differential Galois theory
zbMATH Keywords: Picard-Vessiot theory
zbMATH Keywords: linear differential equations
zbMATH Keywords: differential module
MaRDI profile type: Publication
full work available at URL: j.jsc.2007.07.014
OpenAlex ID: W2035641924
cites work: On rational solutions of systems of linear differential equations
cites work: An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system
cites work: Q4089001
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Connected linear groups as differential Galois groups
cites work: Solving linear differential equations
cites work: Q3998356
cites work: Q5609479
cites work: Lie algebras and Lie groups. 1964 lectures, given at Harvard University.
cites work: Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations
cites work: Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem
cites work: Q4790565
DOI: 10.1016/J.JSC.2007.07.014
description: scientific article; zbMATH DE number 5543812