Galois theory of parameterized differential equations and linear differential algebraic groups

Abstract

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations. * This paper is an expanded version of a talk presented at the conference Singularités deséquations différentielles, systèmes intégrables et groupes quantiques, November 24-27, 2004, Strasbourg, France. The second author would like to thank the organizers of this conference for inviting him.

Projective isomonodromy and Galois groups

“…Note that ∂ x applied to an entry of such an M σ is 0 since these entries are elements of k 0 but that such an entry need not be constant with respect to the elements of Π. One may think of these entries as functions of t. In [8], the authors show that the map σ → M σ is an isomorphism whose image is furthermore a linear differential algebraic group, that is, a group of invertible matrices whose entries satisfies some fixed set of polynomial differential equations (with respect to the derivations Π = {∂ 1 , . .…”

“…Parameterized differential Galois groups (cf. [8], [15]) generalize the concept of differential Galois groups of the classical PicardVessiot theory and we begin this section by briefly describing the underlying theory.…”

“…One can show that the Kolchin closed sets form the closed sets of a topology, called the Kolchin topology on GL n (k 0 ) (cf. [6,7,8,14]). In [8], the authors showed that for parameterized systems of linear differential equations with regular singular points, the parameterized monodromy is Kolchin dense in the PPV-group.…”

“…, t r ), analytic in some domain in C r . A differential Galois theory for such equations was developed in [8] and in greater generality in [15]. Let k 0 be a suitably large field 2 containing C(t 1 , .…”

“…In [8], Cassidy and Singer have developed a new Galois theory for parameterized linear differential equations and defined their parameterized Picard-Vessiot group, PPV-group for short. This is a linear differential algebraic group in the sense of Cassidy [6].…”