# Gorenstein simplices with a given $\delta$-polynomial

## TL;DR

This paper classifies Gorenstein simplices with specific $delta$-polynomials related to prime integer volumes, focusing on cases where the volume is either a square of a prime or a product of two distinct primes.

## Contribution

It provides a complete classification of Gorenstein simplices with certain $delta$-polynomials for prime power and product of primes volumes, extending known results.

## Key findings

- Classified Gorenstein simplices with $delta$-polynomials for $v=p^2$ and $v=pq$
- Determined the number of such simplices up to unimodular equivalence
- Extended the understanding of lattice polytopes with given $delta$-polynomials

## Abstract

To classify the lattice polytopes with a given $\delta$-polynomial is an important open problem in Ehrhart theory. A complete classification of the Gorenstein simplices whose normalized volumes are prime integers is known. In particular, their $\delta$-polynomials are of the form $1+t^k+\cdots+t^{(v-1)k}$, where $k$ and $v$ are positive integers. In the present paper, a complete classification of the Gorenstein simplices with the above $\delta$-polynomials will be performed, when $v$ is either $p^2$ or $pq$, where $p$ and $q$ are prime integers with $p \neq q$. Moreover, we consider the number of Gorenstein simplices, up to unimodular equivalence, with the expected $\delta$-polynomial.