The unreasonable ubiquitousness of quasi-polynomials

TL;DR

This paper explores the widespread occurrence of quasi-polynomials in various mathematical contexts, especially in parametrized sets defined by polynomial inequalities, and proposes a conjecture about their general behavior.

Contribution

It introduces a broad conjecture on the quasi-polynomial nature of parametrized sets defined with logical quantifiers and polynomial inequalities, extending classical results.

Findings

Identifies quasi-polynomial patterns in complex parametrized sets.

Proves several special cases of the conjecture.

Discusses relationships between different conjectural frameworks.

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,...,p_{m-1} such that g(t)=p_i(t) for t=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_1x_1+...+a_dx_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_i 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 of d-tuples of natural numbers 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''.