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arXiv · 2204.13036

Lattice zonotopes of degree 2

Matthias Beck·Ellinor Janssen·Katharina Jochemko
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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 $2$.

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Matthias Beck, Ellinor Janssen, Katharina Jochemko. 2022-04-27. Lattice zonotopes of degree 2. https://arxiv.org/abs/2204.13036

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