The Galois groupoid of \(P_{1}\) and its irreducibility
type: Item
instance of: scholarly article
title: The Galois groupoid of \(P_{1}\) and its irreducibility
zbMATH Open document ID: 1163.34060
DOI: 10.4171/CMH/133
author: Guy Casale
published in: Commentarii Mathematici Helvetici
publication date: 2008-08-13T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: The author addresses the irreducibility of the Galois groupoid of Painlevé P1 equation \[ y'' = 6 y^2 + x. \] The Galois groupoid is defined according to the theory of Malgrange, and the first section of the paper is devoted to a swift and efficient presentation of that theory. Then, the machinery is applied to the foliation defined by P1 on \(\mathbb{C}^3\), and a notion of irreducibility is defined, motivated and proved. The relation with old attacks on the subject (in particular by Poincaré and Drach) is explained, as well as with other approaches to differential Galois theory (Kolchin, Umemura). Last, the author includes the proof of a theorem of Elie Cartan used in the text.
reviewed by: Jacques Sauloy
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 22A22
zbMATH DE Number: 5311211
zbMATH Keywords: Painleve equations
zbMATH Keywords: Galois groupoid
zbMATH Keywords: Lie pseudogroups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5311211