Decomposing Gorenstein polytopes of large index

Abstract

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.