Unimodular triangulations of sufficiently large dilations

TL;DR

This paper proves that for any integral polytope, sufficiently large dilations always admit a unimodular triangulation, strengthening previous results by showing this holds for all larger dilations beyond a certain point.

Contribution

It establishes that beyond a certain dilation factor, all larger dilations of an integral polytope have unimodular triangulations, resolving a longstanding open problem.

Findings

For every integral polytope, there exists a dilation factor after which all larger dilations admit unimodular triangulations.

The result applies to all sufficiently large dilations, not just some specific ones.

This confirms a conjecture about the universal existence of unimodular triangulations for large dilations.

Abstract

An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in $\mathbb{R}^d$ is a triangulation in which all simplices are integral with volume $1/d!$. A classic result of Knudsen, Mumford, and Waterman states that for every integral polytope $P$, there exists a positive integer $c$ such that $cP$ has a unimodular triangulation. We strengthen this result by showing that for every integral polytope $P$, there exists $c$ such that for every positive integer $c' \ge c$, $c'P$ admits a unimodular triangulation. This answers a longstanding question in the area.