Sylvester 单纯形：三角剖分与 Ehrhart 理论性质
Sylvester simplices: Triangulations and Ehrhart-theoretic aspects
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中文总结 AI 辅助
本文研究 Sylvester 单纯形的三角剖分与 Ehrhart 理论性质，证明其 h* 向量单峰，确定 f* 向量部分值，并验证其在低维的 Ehrhart 魔正性。
中文摘要 AI 辅助
Sylvester 单纯形 $\mathsf{Sylv}_d^k$ 是一个具有恰好 $k$ 个内部格点的 $d$ 维格点单纯形。据猜想，对于任意 $k\geq 1$，在所有具有恰好 $k$ 个内部格点的 $d$ 维格点多面体中，Sylvester 单纯形是体积最大化者。更强地，还猜想它们在所有具有恰好 $k$ 个内部格点的 $d$ 维格点多面体中（按分量）最大化 $h^\ast$-向量。然而，Sylvester 单纯形本身似乎很少被研究。特别是，它们的 Ehrhart 理论性质远未被充分理解。在本文中，我们着手解决这个问题。我们描述了 Sylvester 单纯形的旗、正则和幺模三角剖分，并证明了它们的 $h^\ast$-向量是单峰的。此外，我们明确确定了它们的 $f^\ast$-向量某些分量的值，并证明了它们在维度不超过 $6$ 时是 Ehrhart 魔正性的，但在维度 $7$ 时不是。最后，我们详细列出了维度 $7$ 及以下的 Sylvester 单纯形的 Ehrhart 理论量（格点数、Ehrhart 多项式、局部和边界 $h^\ast$-向量、$f^\ast$-向量）的表格。
英文摘要
The Sylvester simplex $\mathsf{Sylv}_d^k$ is a $d$-dimensional lattice simplex with exactly $k$ interior lattice points. Sylvester simplices are conjectured to be the volume maximizers among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points for any $k\geq 1$. Even stronger, it is conjectured that they maximize (entry-wise) the $h^\ast$-vector among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points. Yet, Sylvester simplices seem to be rarely studied in their own right. In particular, their Ehrhart-theoretic properties are far from being well understood. In the present article, we tackle this problem. We describe flag, regular and unimodular triangulations for the Sylvester simplices, and prove that their $h^\ast$-vectors are unimodal. Moreover, we explicitly determine the values of some entries of their $f^\ast$-vectors, and prove that they are Ehrhart magic positive up to dimension $6$ but not in dimension $7$. We conclude by detailing tables of Ehrhart-theoretic quantities (numbers of lattice points, Ehrhart polynomials, local and boundary $h^\ast$-vectors, $f^\ast$-vectors) for Sylvester simplices of dimensions 7 and lower.
