Lattice points in polytopes, box splines, and Todd operators

TL;DR

This paper provides an explicit construction of functions on lattice points in zonotopes using Todd operators, extending previous work and generalizing the Khovanskii-Pukhlikov formula relating volume and lattice points.

Contribution

It introduces an explicit solution to a lattice point interpolation problem using Todd operators, expanding the theoretical framework for lattice point enumeration.

Findings

Explicit solution to interpolation problem using Todd operators

Generalization of Khovanskii-Pukhlikov formula

Connection between differential operators and lattice point enumeration

Abstract

Let $X$ be a list of vectors that is totally unimodular. In a previous article the author proved that every real-valued function on the set of interior lattice points of the zonotope defined by $X$ can be extended to a function on the whole zonotope of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential operator that is contained in the so-called internal $\Pcal$-space. In this paper we construct an explicit solution to this interpolation problem in terms of Todd operators. As a corollary we obtain a slight generalisation of the Khovanskii-Pukhlikov formula that relates the volume and the number of integer points in a smooth lattice polytope.