Liouvillian and algebraic solutions of second and third order linear differential equations
type: Item
instance of: scholarly article
title: Liouvillian and algebraic solutions of second and third order linear differential equations
zbMATH Open document ID: 0802.12005
author: Felix Ulmer
author: Michael F. Singer
published in: Journal of Symbolic Computation
publication date: 1994-12-15T00:00:00Z
review text: Let \(F\) be an ordinary differential field of characteristic 0 and \(L\in F\langle y\rangle\) be a linear homogeneous polynomial. How can one find the Liouvillian solutions of \(L(y)=0\)? In the paper this problem is reduced to the problems of (1) factorization and (2) finding solutions \(u\) such that \(u'/u \in F\) of \(L\) and some polynomials associated with it (symmetric powers of \(L\)). Now there are the algorithms for the solution of the last problems for \(F= \mathbb{Q}(x)\) [see \textit{D. Yu. Grigor'ev}, J. Symb. Comput. 10, 7-37 (1990; Zbl 0728.68067) and \textit{M. F. Singer}, Am. J. Math. 103, 661-682 (1981; Zbl 0477.12026)]. For polynomials \(L\) of the second and third order the authors provide full investigation of the most difficult case when the solution \(u\) of \(L(y)=0\) is algebraic. They show that one can compute the minimal polynomial \(P(y)\in F[y]\) of \(u\). We note that the authors essentially used the tools of representation theory, invariant theory and computer algebra.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 12H20
zbMATH DE Number: 627603
zbMATH Keywords: algebraic solutions
zbMATH Keywords: linear differential equations
zbMATH Keywords: ordinary differential field
zbMATH Keywords: Liouvillian solutions
zbMATH Keywords: factorization
reviewed by: Mykola Grygorenko
describes a project that uses: AXIOM
MaRDI profile type: Publication
OpenAlex ID: W2086624608
DOI: 10.1006/JSCO.1993.1033
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description: scientific article; zbMATH DE number 627603