Algebraic and Algorithmic Aspects of Difference Equations∗

In this course, I will give an elementary introduction to the Galois theory of linear difference equations. This theory shows how to associate a group of matrices with a linear difference equation and shows how group theory can be used to determine properties of the solutions of the equations. I will 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)2 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 ingredients 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 statements of results rather than giving complete proofs. 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. A copy of these notes as well as some related papers can be found at www4.ncsu.edu/∼singer. Click on the link “CIMPA” under other “Other Links”. ∗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, 2012. The author would like to thank the organizers of this school for inviting him. †North Carolina State University, Department of Mathematics, Box 8205, Raleigh, North Carolina 27695-8205, USA, [email protected]. The author was partially supported by NSF Grant CCF-1017217.