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.

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• A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function
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• Approche galoisienne de la transcendance différentielle
• Computing the Galois group of some parameterized linear differential equation of order two
• Computing the differential Galois group of a parameterized second-order linear differential equation
• On parameterized differential Galois extensions
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
• Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier
• Parameterized Differential Equations over k((t))(x)
• Some applications of the parameterized Picard-Vessiot theory
• A simplified categorical approach to several Galois theories
• Malgrange-Galois groupoid of Painlevé VI equation with parameters
• Differential Galois Theory of Linear Difference Equations