发表机构
Faculty of Mathematics, Otto-von-Guericke University Magdeburg（马格德堡奥托·冯·古里克大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了大指标 Gorenstein 多胞体的自由并分解定理，推广了 Batyrev-Juny 金字塔定理，并应用于弦论 E-多项式的消失性判定，证实了相关猜想。
AI 中文摘要
本文证明了关于 Gorenstein 多胞体的自由并分解定理，显著改进了第二作者与 Batyrev、Borger、Haase、Kretschmer 和 Payne 先前合作工作中的结果。特别地，作为 Batyrev-Juny 格金字塔定理的强推广，它蕴含任何指标大于 (d+2)/2 的 d 维 Gorenstein 多胞体 P 都是 Gorenstein 多胞体的自由并。这里，指标（也称为余次数）是使得 rP 为自反多胞体的伸缩因子 r。作为主要应用，我们证明了 Gorenstein 多胞体 P 的弦论 E-多项式消失当且仅当 P 是薄的（即其局部 h*-多项式消失），否则它具有预期的次数。后者由 Batyrev 和第二作者猜想。一般结果的证明是通过 ChatGPT 5.6 Sol 发现的，而 Batyrev-Juny 猜想的极值情形的证明包含在第一作者的硕士论文中。
英文摘要
In this paper we prove a free-join decomposition theorem on Gorenstein polytopes significantly improving upon results in prior joint work of the second author with Batyrev, Borger, Haase, Kretschmer, and Payne. In particular, as a strong extension of the Batyrev-Juny lattice pyramid theorem it implies that any $d$-dimensional Gorenstein polytope $P$ of index larger than $\frac{d+2}{2}$ is a free join of Gorenstein polytopes. Here, the index (also called codegree) is the dilation factor $r$ such that $rP$ is reflexive. As our main application, we prove that the stringy $E$-polynomial of a Gorenstein polytope $P$ vanishes if and only if $P$ is thin (i.e., its local $h^*$-polynomial vanishes), and it has the expected degree otherwise. The latter was conjectured by Batyrev and the second author. The proofs of the general results were found via ChatGPT 5.6 Sol, while a proof of the extremal case of the Batyrev-Juny conjecture is contained in the master thesis of the first author.
Comments16 pages
