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 S-t depending on one or more integer parameters t, and analyze the behavior of the function f(t) = vertical bar S-t vertical bar. 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.

Publisher

Taylor & Francis

Publication Date

3-16-2022

Publication Title

American Mathematical Monthly

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1080/00029890.2022.2010487

Keywords

Factorization length distribution, Numerical semigroups, Integer

Language

English

Format

text

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