Lattice polytopes with a given $h^*$-polynomial
Lattice polytopes with a given $h^*$-polynomial
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具有给定 $h^*$-多项式的晶格多面体
DOI：
10.1090/conm/423/08072
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
V. Batyrev
中科院分区：
文献类型：
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作者：
V. Batyrev
设$\Delta \subset \R^n$是一个n维格多面体。众所周知，\[ h_{\Delta}^*（t）：=（1-t）^{n+1} \sum_{k \geq 0}| k\Delta \cap \Z^n| t^k \]是具有非负整系数的次数为d \leq n+1$的多项式。设$AGL（n，\Z）$是自然作用在$\R^n$上的仿射积分线性变换群。对于给定的多项式h^* \in \Z[t]$，我们用C_{h^*}（n）$表示$n$维格多面体的AGL（n，\Z）$-等价类的个数，使得h^* = h_{\Delta}^*（t）$。本文证明了$\{C_{h^*}（n）\}_{n \geq 1}$是一个单调递增的序列，最终成为常数.这句话来自一个更一般的组合结果，其证明使用交换代数的方法。
Let $\Delta \subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that \[ h_{\Delta}^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |k\Delta \cap \Z^n| t^k \] is a polynomial of degree $d \leq n+1$ with nonnegative integral coefficients. Let $AGL(n, \Z)$ be the group of affine integral linear transformations which naturally acts on $\R^n$. For a given polynomial $h^* \in \Z[t]$, we denote by $C_{h^*}(n)$ the number $AGL(n, \Z)$-equivalence classes of $n$-dimensional lattice polytopes such that $h^* = h_{\Delta}^*(t)$. In this paper we show that $\{C_{h^*}(n) \}_{n \geq 1}$ is a monotone increasing sequence which eventually becomes constant. This statement follows from a more general combinatorial result whose proof uses methods of commutative algebra.