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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)
DOI: https://doi.org/10.1090/memo/1321
Published electronically: June 24, 2021
Table of Contents
Chapters
- 1. Introduction
- 2. Methods
- 3. Examples
- 4. Dilations and the KMW Theorem
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.
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