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

References in corpus (20)

• h-vectors of Gorenstein polytopes
• An algorithm for the classification of smooth Fano polytopes
• Non-connected toric Hilbert schemes
• On the toric ideal of a matroid
• Ehrhart theory, Modular flow reciprocity, and the Tutte polynomial
• Markov bases for noncommutative Fourier analysis of ranked data
• Unmixed bipartite graphs and sublattices of the Boolean lattices
• Unimodular triangulations of dilated 3-polytopes
• Groebner bases of nested configurations
• Unimodular covers of multiples of polytopes
• Gelfand-Tsetlin polytopes and the integer decomposition property
• Non-normal very ample polytopes - constructions and examples
• The Coloring Ideal and Coloring Complex of a Graph
• A generalization of a theorem of G. K. White
• Ehrhart polynomials and stringy Betti numbers
• Bounds on the Coefficients of Tension and Flow Polynomials
• Root polytopes, triangulations, and the subdivision algebra, II
• Markov degree of the Birkhoff model
• Toric fiber products
• Toric rings and ideals of nested configurations

Cited by in corpus (18)

• Toric ideals associated with gap-free graphs
• Semistable reduction in characteristic 0
• The vanishing cycles of curves in toric surfaces I
• Generating functions and triangulations for lecture hall cones
• Gelfand-Tsetlin polytopes and the integer decomposition property
• The local $h$-polynomial of the edgewise subdivision of the simplex
• Non-normal very ample polytopes - constructions and examples
• 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
• Gorenstein properties and integer decomposition properties of lecture hall polytopes
• Quantum jumps of normal polytopes
• Birkhoff polytopes of different type and the orthant-lattice property
• 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