Classification and counting of Gorenstein simplices with $h^*$-polynomial $1+t^k+\cdots+t^{(v-1)k}$

Akiyoshi Tsuchiya

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arXiv · 2605.11820

Classification and counting of Gorenstein simplices with $h^*$-polynomial $1+t^k+\cdots+t^{(v-1)k}$

Akiyoshi Tsuchiya

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Abstract

Hibi, Yoshida, and the author classified Gorenstein simplices which are not lattice pyramids and whose $h^*$-polynomials are of the form $1+t^k+t^{2k}+\cdots+t^{(v-1)k}$ when $v$ is a prime number or the product of two prime numbers. They also conjectured that, for general $v$, the number of unimodular equivalence classes of such simplices depends only on the divisor lattice of $v$. This paper proves the conjecture by giving a constructive classification of Gorenstein simplices whose $h^*$-polynomials are of this form. More precisely, their unimodular equivalence classes are shown to be parametrized by strict divisor chains in the divisor lattice of $v$ together with certain recursive combinatorial data. As a consequence, an explicit formula for the number of equivalence classes in terms of the divisor lattice of $v$ is obtained.

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BibTeX RIS

Akiyoshi Tsuchiya. 2026-05-12. Classification and counting of Gorenstein simplices with $h^*$-polynomial $1+t^k+\cdots+t^{(v-1)k}$. https://arxiv.org/abs/2605.11820

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