Unimodular triangulations of sufficiently 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 $c^{\prime }\ge c$, $c^{\prime }P$ admits a unimodular triangulation. This answers a longstanding question in the area.

40 pages