## Algebra and Combinatorics Seminar

**Date:** December 1, 2017

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 117

**Speaker:** Carlos Arreche, UT Dallas

**Title:** *Projectively integrable linear difference equations and their Galois groups*

**Abstract:** To a linear difference system S is associated a differential Galois group G that measures the differential-algebraic properties of the solutions. We say S is integrable if its solutions also satisfy a linear differential system of the same order, and we say S is projectively integrable if it becomes integrable “modulo scalars”. When the coefficients of S are in C(x) and the difference operator is either a shift, q-dilation, or Mahler operator, we show that if S is integrable then G is abelian, and if S is projectively integrable then G is solvable. As an application of these results one can show certain generating functions arising in combinatorics satisfy no algebraic differential equations. This is joint work with Michael Singer.