The complexity of generating functions for integer points in polyhedra and beyond

Abstract

Motivated by the formula for the sum of the geometric series, we consider various classes  of sets S �¼ Zd of integer points for which an a priori �glong�h Laurent series or polynomial m�¸S xm can be written as a �gshort�h rational function f (S; x). Examples include the sets of integer points in rational polyhedra, integer semigroups, and Hilbert bases of rational cones, among others. We discuss applications to efficient counting and optimization and open questions.