Galois theory of difference equations / Marius van der Put, Michael F. Singer

... The book is divided into two parts: the algebraic theory and the analytic theory. The first chapter starts with the problems of existence and uniqueness of the Picard-Vessiot (ring) extensions. The proofs of these results are very similar to those obtained in the above-mentioned paper of van der Put (where the Picard-Vessiot extension is essentially given by the Hopf algebra of regular functions over the Galois group considered on the coefficient field). The Galois correspondence is studied, as well as the connection with the Tannakian categories (functor fibre). ... The second part of the book (analytic theory) is devoted to asymptotic theory at the point of infinity of linear difference equations, the relation between the formal solutions and the actual analytical ones and the role played by the Galois group from this point of view. It is similar to the asymptotic theory of linear differential equations (but more involved!); the connection between the Picard-Vessiot theory and the asymptotics is due to J.-P. Ramis and J. Martinet ... (MathSciNet)