A Plethora of Polynomials: A Toolbox for Counting Problems

Abstract

A wide variety of problems in combinatorics and discrete optimization depend on counting the set S of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such sets St depending on one or more integer parameters t, and analyze the behavior of the function . In the examples that we investigate, this function exhibits surprising polynomial-like behavior. We end with two broad theorems detailing settings where this polynomial-like behavior must hold. The plethora of examples illustrates the framework in which this behavior occurs and also gives an intuition for many of the proofs, helping us create a toolbox for counting problems like these.