具有非单峰h*-向量的极丰格多面体
A very ample lattice polytope with a non-unimodal $h^*$-vector
AI总结：
本文构造了一个非单峰h*-向量的极丰格多面体，借助ChatGPT 5.6 Sol找到，其为Lasoń等构造的极丰格多面体的笛卡尔平方，回答了相关问题，IDP格多面体是否具单峰h*-向量仍未解决。
AI中文摘要：
格多面体若对每个足够大的k，其锥内高度为k的每个格点都可表示为k个高度为1的格点之和，则称为极丰的，这是著名的整数分解性质（IDP）的弱化形式。本文给出一个极丰格多面体的例子，其h*-向量为非单峰的，其中h*-向量是该格多面体的Ehrhart级数分子的系数向量。这一结果回答了Ferroni与Higashitani的问题，以及Balletti的相关问题。关于IDP格多面体是否具有单峰h*-向量的核心问题仍未解决。该例子借助ChatGPT 5.6 Sol找到，它正是Lasoń与Michalek构造的一类极丰例子中某格多面体的笛卡尔平方。
英文摘要：
Lattice polytopes are called very ample if for every sufficiently large $k$ every lattice point of height $k$ in the cone over the lattice polytope is the sum of $k$ lattice points of height $1$. This is a weakening of the well-known integer decomposition property (also called IDP). We give an example of a very ample lattice polytope whose $h^*$-vector is non-unimodal. Here, the $h^*$-vector is the coefficient vector of the numerator of the Ehrhart series of the lattice polytope. This answers a question of Ferroni and Higashitani, as well as a related question by Balletti. The main question whether IDP lattice polytopes have unimodal $h^*$-vector is still open. The example was found using ChatGPT 5.6 Sol. It is just the Cartesian square of a lattice polytope belonging to a class of very ample examples constructed by Lasoń and Michalek.
