发表机构
Nankai University; Center for Combinatorics, LPMC（南开大学; 组合数学中心，LPMC）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究格多胞体上八种运算（如笛卡尔积、闵可夫斯基和等）对三类Ehrhart性质（系数正性、$h^*$-多项式系数性质、几何性质）的保持性，通过建立保持定理或构造反例，并探讨非保持情形下的充分或等价条件。
AI 中文摘要
本文研究了格多胞体上各种运算下三类Ehrhart性质的保持性。这些运算包括笛卡尔积、格连接、格金字塔、自由和、闵可夫斯基和、凯莱和、自反极性以及整数膨胀。具体而言，我们考虑了Ehrhart理论中经常研究的以下性质：(i) Ehrhart系数的正性，包括Ehrhart正性和魔幻正性。特别地，我们给出了由这些运算生成的多胞体的格点计数公式。(ii) $h^*$-多项式的系数性质，包括对称性、单峰性、对数凹性、超对数凹性、实根性以及$\gamma$-正性。(iii) 几何性质，包括生成性质、整数分解性质、非常丰沛性以及单模三角剖分、正则单模三角剖分和二次三角剖分的存在性。我们确定了在八种运算下哪些性质被保持，建立了保持定理或构造了显式反例。此外，当性质在一般情况下不被保持时，我们研究了其保持的充分条件或等价条件。
英文摘要
This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. Specifically, we consider the following properties frequently studied in Ehrhart theory: (i): Positivity of Ehrhart coefficients, including Ehrhart positivity and magic positivity. In particular, we present lattice point counting formulas for the polytopes generated by these operations. (ii): Coefficient properties of the $h^*$-polynomial, including symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, and $γ$-positivity. (iii): Geometric properties, including the spanning property, the integer decomposition property, very ampleness, and the existence of unimodular triangulation, regular unimodular triangulation, and quadratic triangulation. We determine which properties are preserved under these eight operations, establishing preservation theorems or constructing explicit counterexamples. Furthermore, when a property is not preserved in general, we investigate sufficient or equivalent conditions for its preservation.
