Algebraic and Algorithmic Aspects of Difference Equations

These are lecture notes for a series of talks given at the CIMPA Research School Galois Theory of Difference Equations held in Santa Marta, Columbia, July 23-August 1, 20121. In this course, I gave an elementary introduction to the Galois theory of linear difference equations from an algebraic and algorithmic perspective. This theory shows how to associate a group of matrices with a linear difference equa-tion and shows how group theory can be used to determine properties of the solutions of the equations. These notes begin by giving an introduction to the theory of linear algebraic groups, those groups that occur as Galois groups. I will then present the basic features of the Galois theory and show how this theory can be used to determine algebraic properties of sequences of numbers determined by linear recurrences. In particular I will show how the Galois theory leads to algorithms to determine algebraic relations among such solutions (such as the relation F (n)F (n+ 2)−F (n+ 1)2 = (−1)n among the Fibonacci numbers F (n)) and algorithms to express such solutions in “finite terms”. The goal of my course and of these notes is to give a taste of the various ingre-dients that are used to build the Galois theory of linear difference equations and the applications of this theory. Therefore I will focus on explaining definitions and state-ments of results rather than giving complete proofs. I will assume that the reader has a basic knowledge of abstract algebra (groups, rings, fields, ideals,...). I hope that these notes give enough knowledge and evoke enough interest that you will go to the sources mentioned and delve further into the subject