Galoisian methods for testing irreducibility of order two nonlinear differential equations

Guy Casale and Jacques-Arthur Weil

Vol. 297 (2018), No. 2, 299–337

DOI: 10.2140/pjm.2018.297.299

Abstract

We provide a criterion to compute the Malgrange pseudogroup, the nonlinear analog of the differential Galois group, for classes of second order differential equations. Let $G_k$ be the differential Galois groups of their $k$-th variational equations along an algebraic solution $\Gamma$. We show that if the dimension of one of the $G_k$ is large enough, then the Malgrange pseudogroup is known. This in turn proves the irreducibility of the original nonlinear differential equation. To make the criterion applicable, we give a method to compute the dimensions of the variational Galois groups $G_k$ via constructive reduced form theory. As an application, we reprove the irreducibility of the second and third Painlevé equations for special values of their parameter. In the appendices, we recast the various notions of variational equations found in the literature and prove their equivalences.

Keywords

ordinary differential equations, differential Galois theory, Painlevé equations, computer algebra

Mathematical Subject Classification 2010

Primary: 34A05, 34A26, 34M03, 34M15, 34M25

Secondary: 17B45, 20G05, 34M55

Milestones

Received: 29 October 2017

Revised: 26 April 2018

Accepted: 30 April 2018

Published: 20 December 2018