精制Ehrhart系列和加强型环

Pub Date : 2023-10-24 DOI:10.1556/012.2023.01541

Praise Adeyemo, Balázs Szendrői

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引用次数: 0

摘要

我们研究了由Chapoton首先考虑的闭多面体的Ehrhart级数的自然精化集。我们用交换代数计算了维数为𝑑的单纯形、维数为𝑑的交叉多面体和维数为𝑑≤3的超立方体的完全一般精炼级数。我们用不同的参数推导出𝑞-integers的乘积的求和公式，推广了MacMahon和Carlitz的经典恒等式。我们也给出了某一精化欧拉多项式的代数性质。

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Refined Ehrhart Series and Bigraded Rings

We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension 𝑑, a cross-polytope of dimension 𝑑, respectively a hypercube of dimension 𝑑 ≤ 3, using commutative algebra. We deduce summation formulae for products of 𝑞-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.

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