Lattice zonotopes of degree 2

TL;DR

This paper classifies all Ehrhart polynomials of 3-dimensional lattice zonotopes with degree 2, providing a constructive characterization using solid angles and lattice width, thus advancing the understanding of Ehrhart theory.

Contribution

It offers a complete classification of Ehrhart polynomials for degree 2 lattice zonotopes, complementing prior partial results and providing a constructive proof.

Findings

All Ehrhart polynomials of 3D degree 2 lattice zonotopes are characterized.

A constructive method using solid angles and lattice width is developed.

The classification extends previous work by Scott, Treutlein, and Henk-Tagami.

Abstract

The Ehrhart polynomial $ehr_P (n)$ of a lattice polytope $P$ gives the number of integer lattice points in the $n$-th dilate of $P$ for all integers $n\geq 0$. The degree of $P$ is defined as the degree of its $h^\ast$-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree $2$ thereby complementing results of Scott (1976), Treutlein (2010), and Henk-Tagami (2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all $3$-dimensional zonotopes of degree $2$.