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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2206.04175 |
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| _version_ | 1866909322371399680 |
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| author | Bajo, Esme Beck, Matthias |
| author_facet | Bajo, Esme Beck, Matthias |
| contents | If $P$ is a lattice polytope (i.e., $P$ is the convex hull of finitely many integer points in $\mathbb{R}^d$) of dimension $d$, Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|nP \cap \mathbb{Z}^d|$ is a degree-$d$ polynomial in the integer variable $n$. Equivalently, the generating function $1 + \sum_{n\geq 1} |nP \cap \mathbb{Z}^d| \, z^n$ is a rational function of the form $\frac{ h^\ast(z) }{ (1-z)^{ d+1 } }$; we call $h^\ast(z)$ the $h^\ast$-polynomial of $P$. There are several known necessary conditions for $h^\ast$-polynomials, including results by Hibi (1990), Stanley (1991), and Stapledon (2009), who used an interplay of arithmetic (integer-point structure) and topological (local $h$-vectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the $h^\ast$-polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. We include applications for (rational) Gorenstein polytopes and rational Ehrhart dilations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04175 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Boundary $h^\ast$-polynomials of rational polytopes Bajo, Esme Beck, Matthias Combinatorics If $P$ is a lattice polytope (i.e., $P$ is the convex hull of finitely many integer points in $\mathbb{R}^d$) of dimension $d$, Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|nP \cap \mathbb{Z}^d|$ is a degree-$d$ polynomial in the integer variable $n$. Equivalently, the generating function $1 + \sum_{n\geq 1} |nP \cap \mathbb{Z}^d| \, z^n$ is a rational function of the form $\frac{ h^\ast(z) }{ (1-z)^{ d+1 } }$; we call $h^\ast(z)$ the $h^\ast$-polynomial of $P$. There are several known necessary conditions for $h^\ast$-polynomials, including results by Hibi (1990), Stanley (1991), and Stapledon (2009), who used an interplay of arithmetic (integer-point structure) and topological (local $h$-vectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the $h^\ast$-polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. We include applications for (rational) Gorenstein polytopes and rational Ehrhart dilations. |
| title | Boundary $h^\ast$-polynomials of rational polytopes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2206.04175 |