The unreasonable ubiquitousness of quasi-polynomials

Abstract

A function g, with domain the natural numbers, is a quasi-polynomial if there exists a periodmand polynomials p0, p1, . . . , pm-1 such that g(t) = pi(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 a1x1 + . . . + adxd ≤ b(t). Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the ai 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 St that are defined with quantifiers (8, 9), boolean operations (and, or, not), and statements of the form a1(t)x1 + . . . + ad(t)xd ≤ b(t), where ai(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. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS).