Irreducible linear differential equations of prime order
type: Item
instance of: scholarly article
title: Irreducible linear differential equations of prime order
zbMATH Open document ID: 0823.12003
author: Felix Ulmer
published in: Journal of Symbolic Computation
publication date: 1995-07-13T00:00:00Z
review text: Let \(L(y)= \sum_{i=0}^ p a_ i (d^ iy/ dx^ i) =0\) be an irreducible differential equation of prime order where \(a_ i\) lie in a differential field \(k\) with an algebraically closed field of constants. Suppose the differential Galois group of \(L(v) =0\) is unimodular, and \({\mathcal P}_ R= \{P_ 1, P_ 2, \ldots\}\) is the set of all minimal polynomials of an algebraic solution of minimal degree of the associated Riccati equation \(R(u) =0\). In case the differential Galois group \({\mathcal G} (L)\) of \(L(y)=0\) is an imprimitive subgroup of \(\text{SL} (p, \mathbb{C})\), then (1) \({\mathcal P}_ R\) is non empty and consists of polynomials of degree \(p\). (2) If \({\mathcal G} (L)\) has no faithful central monomial representation of degree \(p\), then \({\mathcal P}_ R\) contains at most \((p+1)\) elements and there is a bijection between maximal normal abelian subgroups of \({\mathcal G} (L)\) and elements of \({\mathcal P}_ R\). (3) If \({\mathcal G} (L)\) has no faithful central monomial representation of degree \(p\) and is not isomorphic to a split extension of a cyclic group \(C\) of order dividing \(p-1\) and \(N_ p\) which is the unique (up to isomorphism) non abelian subgroup of order \(p\) in \(\text{SL} (p, \mathbb{C})\), then \({\mathcal P}_ R\) contains exactly one element. If \(L(y) =0\) has a Liouvillian solution over \(k\), then either (or both) of the following holds namely, \({\mathcal P}_ R\) contains exactly one element or \({\mathcal G} (L)\) belongs to a finite set of finite subgroups of \(\text{SL} (p, \mathbb{C})\). In the latter case all solutions of \(L(y) =0\) must be algebraic and we can use the method of the author and \textit{M. Singer} [J. Symb. Comput. 16, 37-73 (1993; Zbl 0802.12005)] to compute the minimal polynomial of a solution of \(L(Y) =0\). The author illustrates the method by working out examples of second and third order equations. Finally, necessary conditions are given for imprimitive finite subgroups which exclude c…
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 762069
zbMATH Keywords: irreducible differential equation of prime order
zbMATH Keywords: differential field
zbMATH Keywords: differential Galois group
zbMATH Keywords: Liouvillian solution
reviewed by: N. Sankaran
MaRDI profile type: Publication
OpenAlex ID: W2046968295
DOI: 10.1006/JSCO.1994.1055
description: scientific article; zbMATH DE number 762069