# Existence of regular unimodular triangulations of dilated empty simplices

## TL;DR

This paper establishes the precise conditions under which the dilated versions of certain empty simplices admit regular unimodular triangulations, advancing understanding in geometric combinatorics.

## Contribution

It provides a complete characterization of when the k-th dilation of specific empty simplices has a regular unimodular triangulation.

## Key findings

- Necessary and sufficient conditions for regular unimodular triangulations of dilated simplices.
- Characterization based on the $oldsymbol{ ext{delta}}$-polynomial form.
- Extension of prior results on empty simplices and their triangulations.

## Abstract

Given integers $k$ and $m$ with $k \geq 2$ and $m \geq 2$, let $P$ be an empty simplex of dimension $(2k-1)$ whose $\delta$-polynomial is of the form $1+(m-1)t^k$. In the present paper, the necessary and sufficient condition for the $k$-th dilation $kP$ of $P$ to have a regular unimodular triangulation will be presented.