Computing parametric weighted Ehrhart polynomials of smooth polytopes

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.