论文标题
边界$ h^\ ast $ - 理性多型
Boundary $h^\ast$-polynomials of rational polytopes
论文作者
论文摘要
如果$ p $是晶格多层（即，$ p $是$ \ mathbb {r}^d $ dimension $ d $的有限整数点的凸形船体，ehrhart的著名定理（1962）声称，整数计数函数$ | np \ cap $ cap $ | $ cap $ cap $ cap $ | $ |整数变量$ n $。同等地，生成函数$ 1 + \ sum_ {n \ geq 1} | np \ cap \ mathbb {z}^d | \，z^n $是$ \ frac {h^\ ast（z）} {（1-z）^{d+1}} $;我们称$ h^\ ast（z）$ $ h^\ ast $ -polynomial的$ p $。 $ h^\ ast $ - 多元素的几个必要条件，包括Hibi（1990），Stanley（1991）和Stapledon（2009）的结果，他们使用了算术（整数点结构）的相互作用（整数 - 点结构）和拓扑（本地$ h $ h $ triangulations of Triangulations of Triangulations of Triangulations of Triangulations）数据的数据。我们引入了另一种ANSATZ，通过$ h^\ ast $ - 多层元素的$ h^\ ast $多态度来理解ehrhart理论，以统一的方式恢复了上述所有结果及其对理性多面体的扩展。我们包括（理性）Gorenstein多面体和合理的Ehrhart扩张的应用。
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.
