Periods of Ehrhart coefficients of rational polytopes

TL;DR

The paper investigates which Ehrhart quasi-polynomials can arise from rational polytopes by focusing on the periods of the coefficient functions. It develops a constructive approach using two-dimensional building blocks with the complement property $\operatorname{ehr}_{P}(x) \equiv -\operatorname{ehr}_{\ell}(x)$ and dimension-raising via $i$-fold pyramids to realize prescribed period sequences. The authors prove that for any $p \ge 1$ there exists an $n$-dimensional convex rational polytope with period sequence $(1, p, 1, \dots, 1)$, and, conditionally on solutions to the ideal PTE problem, that there exist $n$-dimensional nonconvex polytopes with period sequence $(1, \dots, 1, p, 1)$ in dimensions $3 \le n \le 11$ or $n = 13$. These constructions extend the landscape of realizable period sequences beyond McMullen’s bounds and illustrate how both convex and nonconvex geometries contribute to the Ehrhart theory of rational polytopes.

Abstract

Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ -- that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.

Figures (1)

• Figure 1: The polytope $H_{3}$ in the case where $p = 2$.

Theorems & Definitions (12)

• Theorem 1.1: McMullen McM1978
• Theorem 1.2: Beck et al. BecSamWoo2008
• Theorem 1.3
• Theorem 1.4
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof