Ehrhart quasi-polynomials and parallel translations

2023/07/16 by Higashitani, Akihiro, Murai, Satoshi, Yoshinaga, Masahiko · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.08151

Abstract

Given a rational polytope P ⊂ \mathbb R^d, the numerical function counting lattice points in the integral dilations of P is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial ehr_P of P. In this paper we study the following problem: Given a rational d-polytope P ⊂ \mathbb R^d, is there a nice way to know Ehrhart quasi-polynomials of translated polytopes P+ \mathbf v for all \mathbf v ∈ \mathbb Q^d? We provide a way to compute such Ehrhart quasi-polynomials using a certain toric arrangement and lattice point counting functions of translated cones of P. This method allows us to visualize how constituent polynomials of ehr_{P+\mathbf v} change in the torus \mathbb R^d/\mathbb Z^d. We also prove that information of ehr_{P+\mathbf v} for all \mathbf v ∈ \mathbb Q^d determines the rational d-polytope P ⊂ \mathbb R^d up to translations by integer vectors, and characterize all rational d-polytopes P ⊂ \mathbb R^d such that ehr_{P+\mathbf v} is symmetric for all \mathbf v ∈ \mathbb Q^d.

Cited by

• Ehrhart Theory over Abelian Group Rings

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