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arXiv math/0602593math.COmath.ACmath.AG
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具有给定$h^*$ -多项式的格多面体

Lattice polytopes with a given $h^*$-polynomial

Victor Batyrev

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10 pages. AMS-LaTeX, some typos were corrected
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摘要：

设$Δ\subset \R^n$是$n$维格多面体.众所周知，$h_Δ^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |kΔ\cap \Z^n| t^k $是具有非负整系数的$d \leq n$次多项式。设$AGL(n, \Z)$是自然作用于$\R^n$的可逆仿射积分变换群。对于给定的多项式$h^* \in \Z[t]$，我们用$C_{h^*}(n)$表示$n$维格多面体的个数$AGL(n, \Z)$ -等价类，使得$h^* = h_Δ^*(t)$ .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。

英文摘要：

Let $Δ\subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_Δ^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |kΔ\cap \Z^n| t^k $ is a polynomial of degree $d \leq n$ with nonnegative integral coefficients. Let $AGL(n, \Z)$ be the group of invertible affine integral 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 $h^* = h_Δ^*(t)$. 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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