Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds

Abstract

We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a rational polytope. There is an interesting appearance of an extra complex z-parameter, which seems to allow for more compact formulations. We also give similar formulas for discrete moments of rational polytopes, and their positive dilates, objects known in the literature as sums of polynomials over a polytope. The appearance of the Barnes polynomials and the Barnes numbers allow for explicit computations. From this work, it is clear that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds. These discrete moments are in general summed over a particular lattice flow on a compact torus, defined in this paper. Some of the consequences involve novel vanishing identities for rational polytopes. As another consequence, we obtain a differential equation for discrete moments of rational polytopes, which extends the work of Eva Linke. For smooth polytopes, we obtain novel and much simpler formulations of Ehrhart polynomials, discrete moments, and vanishing identities that may be of independent interest from the perspective of Barnes polynomials and Barnes numbers. These formulations show the utility of Barnes polynomials in geometric combinatorics, due to their very rich structure that extends the 1-dimensional Bernoulli polynomials.