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Existence of unimodular triangulations — positive results

About this Title

Christian Haase, Andreas Paffenholz, Lindsey C. Piechnik and Francisco Santos

Publication: Memoirs of the American Mathematical Society
Publication Year: 2021; Volume 270, Number 1321
ISBNs: 978-1-4704-4716-8 (print); 978-1-4704-6530-8 (online)
Published electronically: June 24, 2021

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Table of Contents


  • 1. Introduction
  • 2. Methods
  • 3. Examples
  • 4. Dilations and the KMW Theorem


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.

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