Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups

TL;DR

This paper develops a Galois theory for parameterized linear differential equations, where Galois groups are linear differential algebraic groups, providing new tools for classifying and understanding such systems.

Contribution

It introduces a Galois theory framework for parameterized differential equations with differential algebraic group Galois groups, linking group theory and differential equations.

Findings

Established basic constructions and results of the theory

Provided examples illustrating the theory

Connected isomonodromic families to the Galois framework

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.