Lattice zonotopes of degree 2
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914954928455680 |
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| author | Beck, Matthias Janssen, Ellinor Jochemko, Katharina |
| author_facet | Beck, Matthias Janssen, Ellinor Jochemko, Katharina |
| contents | 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 $2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_13036 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lattice zonotopes of degree 2 Beck, Matthias Janssen, Ellinor Jochemko, Katharina Combinatorics Primary: 52B20, Secondary: 05A15, 11H06 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 $2$. |
| title | Lattice zonotopes of degree 2 |
| topic | Combinatorics Primary: 52B20, Secondary: 05A15, 11H06 |
| url | https://arxiv.org/abs/2204.13036 |