Splines, lattice points, and arithmetic matroids

TL;DR

This paper improves the Brion-Vergne formula for counting lattice points in polytopes defined by a matrix and explores related differential operator spaces, linking them to arithmetic matroids and zonotopal algebra.

Contribution

It refines the Brion-Vergne formula and studies finite-dimensional differential operator spaces connected to arithmetic matroids and zonotopal algebra.

Findings

Improved the Brion-Vergne formula for lattice point enumeration.

Identified finite-dimensional spaces of differential operators related to the problem.

Connected these spaces to the Tutte polynomial of arithmetic matroids and zonotopal algebra.

Abstract

Let $X$ be a $(d\times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \{w \ge 0 : X w = u \}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^d$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. The Brion-Vergne formula implies that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this article we slightly improve the Brion-Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate $T_X$) and the space of nice differential operators (i.e. operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix $X$. They are closely related to the $\mathcal{P}$-spaces studied by Ardila-Postnikov and Holtz-Ron in the context of zonotopal algebra and power ideals.