Galois Theory of Difference Equations

This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.

Keywords: Difference Equations, Galois Theory, Connection Matrices, Multisummation.

Contents: Introduction.- Algebraic Theory,- Picard-Vessiot Rings.- Algorithms for Difference Equation.- The Inverse Problem for Difference Equations.- The Ring S of Sequences.- An Excursion in Positive Characteristic.- Difference Modules over P.- Classification and Canonical Forms.- Semi-regular Difference Equations.- Mild Difference Equations.- Examples of Equations and Galois Groups.- Wild Difference Equations.- q-Difference Equations.- Bibliography.- Index.- Notations.