Computing parametric weighted Ehrhart polynomials of smooth polytopes

TL;DR

This paper develops an algorithm to compute parametric weighted Ehrhart polynomials for smooth polytopes, revealing their piecewise polynomial nature as the polytopes deform while maintaining facet orientations.

Contribution

It introduces a new method and implementation for calculating weighted Ehrhart and h*-polynomials for smooth polytopes, utilizing a weighted Euler-Maclaurin formula.

Findings

Coefficients are piecewise polynomial functions under polytope deformations.

Algorithm successfully computes these polynomials for type A alcoved polytopes.

Discussion on the signs of weighted h*-polynomial coefficients.

Abstract

We show that when integral polytopes are deformed while keeping the same facet normal vectors, the coefficients of weighted Ehrhart and $h^*$-polynomials are piecewise polynomial functions in the ``right hand sides'' of the linear inequalities defining the polytopes. We give an algorithm and an implementation in SageMath for computing these polynomials for smooth polytopes, such as type $A$ alcoved polytopes, using a weighted Euler-Maclaurin type formula by Khovanski\v{i} and Pukhlikov. We discuss some natural questions concerning signs of the coefficients of the weighted $h^*$-polynomials.