Abstract
A function $g$, with domain the natural numbers, is a quasi-polynomial if there exists a period $m$ and polynomials $p_0,p_1,\ldots,p_m-1$ such that $g(t)=p_i(t)$ for $t\equiv i\bmod m$. 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 $S_t$ that are defined with quantifiers $(\forall , ∃)$, 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.
Highlights
Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the ai are allowed to vary with t
We conjecture that any family of sets St – defined with quantifiers (∀, ∃), boolean operations, and statements of the form a(t) · x ≤ b(t) (where a(t) ∈ Q[t]d, b(t) ∈ Q[t], and · is the standard dot product) – exhibits eventual quasi-polynomial behavior, as well as rational generating function behavior
Sturmfels (1995) effectively proved this generalization of Ehrhart theory: Theorem 6 Let St be the set of integer points, x ∈ Zd, in a polyhedron defined with linear inequalities a · x ≤ b(t), where a ∈ Zd and b(t) is a degree 1 polynomial in Z[t]
Summary
We survey classical appearances of quasi-polynomials (though Section 1.3 might be new even to readers already familiar with Ehrhart theory). We conjecture that any family of sets St – defined with quantifiers (∀, ∃), boolean operations (and, or, not), and statements of the form a(t) · x ≤ b(t) (where a(t) ∈ Q[t]d, b(t) ∈ Q[t], and · is the standard dot product) – exhibits eventual quasi-polynomial behavior, as well as rational generating function behavior. T 2 t+1 if t even, if t odd, is a quasi-polynomial with period 2. This example makes it clear that the ubiquitousness of quasi-polynomials shouldn’t be too surprising: anywhere there are floor functions, quasi-polynomials are likely to appear. Note that Example 2 demonstrates that such quasi-polynomials may still require rational coefficients
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More From: Discrete Mathematics & Theoretical Computer Science
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