AI 中文总结
本文研究交叉多面体闵可夫斯基和的格点计数，引入支撑计数函数，利用根多面体对称性证明相关猜想，导出格点数量公式并推导Ehrhart多项式等相关结果。
AI 中文摘要
受Postnikov关于单形闵可夫斯基和中格点计数研究的启发，我们研究交叉多面体闵可夫斯基和中的格点，并建立了类似的结果及若干相关推论。特别地，我们引入与Postnikov提出的 draconian 序列概念相关的支撑计数函数，并证明其与对应根多面体的h*-多项式一致，这为h*-多项式提供了新的解释，也为计算对应多面体的体积提供了简便方法。我们进一步利用此类根多面体的对称性，建立支撑计数函数的对偶性质，进而证明了Chapoton与Athanasiadis关于预序的h-多项式的猜想。作为直接推论，我们证明交叉多面体与其对偶多面体的闵可夫斯基和具有相同数量的格点。该对偶性随后导出了基于draconian序列的交叉多面体闵可夫斯基和的格点数量通用公式。此公式使我们能够计算这些多面体的Ehrhart多项式，并证明它们满足Ehrhart正性；此外，该公式还可推导出其边界格点数量及表面体积的类似公式。
英文摘要
Motivated by Postnikov's study of lattice-point enumeration in Minkowski sums of simplices, we investigate lattice points in Minkowski sums of cross polytopes and establish analogous results, together with several related consequences. In particular, we introduce the support-enumerator associated with Postnikov's notion of draconian sequences and show that it coincides with the $h^*$-polynomial of the corresponding root polytope. This provides a new interpretation of the $h^*$-polynomial and yields a simple method for computing the volume of the corresponding polytope. By exploiting the symmetry of these root polytopes, we further establish a duality property for support-enumerators, which in turn provides a proof of a conjecture by Athanasiadis and Chapoton concerning the $h$-polynomials of preorders. Consequently, we obtain a formula for the number of lattice points in Minkowski sums of cross polytopes in terms of draconian sequences and show that these polytopes are Ehrhart positive. Furthermore, this formula leads to analogous expressions for the number of lattice points on their boundaries and for their surface volumes.
Comments25 pages. We improved the exposition of the paper
