The unreasonable ubiquitousness of quasi-polynomials

Abstract

A function g, with domain the natural numbers, is a quasi-polynomial if there exists a period m and polynomials p(0),p(1),...,Pm-1 such that g(t) - p(i)(t) for t equivalent to i mod m. Quasi-polynomials classically - and "reasonably" - appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable t, and defined by linear inequalities of the form a(1)x(1) +...+a(d)x(d) <= b(t). Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the a, are also allowed to vary with t. We discuss these "unreasonable" results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets S-t subset of N-d that are defined with quantifiers (for all, there exists), boolean operations (and, or, not), and statements of the form a(1)(t)x(1)+...+a(d)(t)x(d) <= b(t), where a(i)(t) and b(t) are polynomials in t. These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures. The title is a play on Eugene Wigner's "The unreasonable effectiveness of mathematics in the natural sciences".

Publisher

Electronic Journal Of Combinatorics

Publication Date

2-28-2014

Publication Title

Electronic Journal Of Combinatorics

Department

Mathematics

Document Type

Article

Keywords

Ehrhart polynomials, Generating functions, Presburger arithmetic, Quasi-plynomials, Rational generating functions

Language

English

Format

text

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