On Ramis's solution of the local inverse problem of differential Galois theory
type: Item
instance of: scholarly article
title: On Ramis's solution of the local inverse problem of differential Galois theory
zbMATH Open document ID: 0854.12001
DOI: 10.1016/0022-4049(95)00104-2
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-19T00:00:00Z
review text: \textit{J. P. Ramis} [About the inverse problem in differential Galois theory: The differential Abhyankar conjecture. In: Braaksma, B. L. J. (ed.) et al., The Stokes phenomenon and Hilbert's 16th problem. Proceedings of the workshop, Groningen, 261-278 (1996; Zbl 0860.12003)] gave necessary and sufficient conditions for a linear algebraic group to be the Galois group of a Picard-Vessiot extension of the field \(C\{ x\}[ x^{-1}]\) of germs of meromorphic functions at zero. In this paper, the authors give equivalent simple group theoretic conditions. The main result is the following: Theorem. Let \(G\) be a linear algebraic group. The following statements are equivalent: (1) \(G\) is the Galois group of some Picard-Vessiot extension of \(C\{ x\}[ x^{-1} ]\). (2) The following three conditions hold: (a) \(G/ G^0\) is cyclic, (b) the dimension of \(R_u/[ R_u, G^0]\) is at most 1, (3) \(G/ G^0\) acts trivially on \(R_u/[ R_u, G^0]\).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 896571
zbMATH Keywords: Galois group
zbMATH Keywords: Picard-Vessiot extension
zbMATH Keywords: linear algebraic group
author: Claude Mitschi
author: Michael F. Singer
reviewed by: Mykola Grygorenko
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00104-2
OpenAlex ID: W2170956753
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description: scientific article; zbMATH DE number 896571