Existence of unimodular triangulations - positive results

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.

89 pages; changes from v2 and v1: the survey part has been expanded, in particular the section on open questions

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• Amoebas of curves and the Lyashko-Looijenga map
• Log smoothness and polystability over valuation rings
• Ehrhart-equivalent $3$-polytopes are equidecomposable
• Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes
• Stapledon Decompositions and Inequalities for Coefficients of Chromatic Polynomials
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• Quantum jumps of normal polytopes
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• Unimodular covers of 3-dimensional parallelepipeds and Cayley sums
• Ehrhart-Equivalence, Equidecomposability, and Unimodular Equivalence of Integral Polytopes
• Unimodular triangulations of simplicial cones by short vectors