Existence of unimodular triangulations - positive results

TL;DR

This paper reviews classes of lattice polytopes with unimodular triangulations, provides constructions preserving their existence, and offers an explicit bound for dilations ensuring such triangulations, advancing understanding in algebraic geometry and combinatorics.

Contribution

It presents the first effective proof that every lattice polytope has a dilation with a unimodular triangulation, including an explicit bound for the dilation factor.

Findings

Every lattice polytope has a dilation admitting a unimodular triangulation.

Provided an explicit, doubly exponential bound for the dilation factor.

Reviewed classes of polytopes with unimodular triangulations and construction methods.

Abstract

Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course, combinatorics. In this article, we review several classes of polytopes that do have unimodular triangulations and constructions that preserve their existence. We include, in particular, the first effective proof of the classical result by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an explicit (although doubly exponential) bound for the dilation factor.