A density theorem in parametrized differential Galois theory
type: Item
instance of: scholarly article
title: A density theorem in parametrized differential Galois theory
zbMATH Open document ID: 1364.12004
author: Thomas Dreyfus
published in: Pacific Journal of Mathematics
publication date: 2014-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.2904
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34M15
Mathematics Subject Classification ID: 34M03
zbMATH DE Number: 6369630
zbMATH Keywords: parametrized differential Galois theory
zbMATH Keywords: Stokes phenomenon
zbMATH Keywords: inverse problem
zbMATH Keywords: complete integrability
zbMATH Keywords: isomonodromy
reviewed by: Mykola Grygorenko
MaRDI profile type: Publication
OpenAlex ID: W3102026983
arXiv ID: 1203.2904
DOI: 10.2140/PJM.2014.271.87
review text: Let \(F\) be a differential field of characteristic zero with an algebraically closed field of constants \(C\) and let \(t_1,\ldots ,t_n\) be constants transcendental over \(C\). In the framework of the differential Galois theory developed by Kolchin, it is difficult to investigate differential equations of the form NEWLINE\[NEWLINE\delta y=Ay,\quad A\in M(m,F(t_1,\ldots ,t_n)).NEWLINE\]NEWLINE Because most of the constructive results of the theory are obtained in the case of a closed field of constants. For example, it is difficult to prove the existence of the Picard-Vessiot extension of the field \(\mathbb{C}(\rho(z),\rho'(z),\lambda)\) corresponding to the Lamé equation \(y''-n(n+1)\rho(z)y=\lambda y\), when \(\lambda\) is a transcendental over \(\mathbb C\) constant (parameter). At the same time, in applied problems, it is often necessary to investigate equations with parameters. In 2007, \textit{P. J. Cassidy} and \textit{M. F. Singer} [in: Differential equations and quantum groups. Andrey A. Bolibrukh memorial volume. Zürich: European Mathematical Society Publishing House, 113--155 (2007; Zbl 1230.12003)] presented a special Galois theory for linear differential equations with parameters, similar to Kolchin's theory for equations without parameters. In particular, they showed that the role of weakly singular points in the new theory is the same as in the Kolchin theory. In this article, the author proves a stronger resultNEWLINENEWLINETheorem 2.20 (parametrized analog of the density theorem of Ramis). The group generated by the parametrized monodromy, the parametrized exponential torus and the parametrized Stokes operators is dense in the parametrized differential Galois group for the Kolchin topology.NEWLINENEWLINEA number of other interesting results on the inverse problem, isomonodromy and complete integrability for equations with parameters are also obtained in the article.
description: scientific article; zbMATH DE number 6369630