Existence of Unimodular Triangulations-Positive Results.

Format:

Book

Author/Creator:

Haase, Christian.

Contributor:

Paffenholz, Andreas.

Piechnik, Lindsay C.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.270

Language:

English

Subjects (All):

Convex polytopes.

Triangularization (Mathematics).

Combinatorial geometry.

Physical Description:

1 online resource (96 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"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"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. What?

1.2. Why? Who?

1.3. What is new?

1.4. What is not here

1.5. What is left?

Chapter 2. Methods

2.1. Pulling Triangulations

2.2. Push-forward subdivisions and pull-back subdivisions

2.3. Joins and (Fiber) Products

2.4. Toric Gröbner Bases

Chapter 3. Examples

3.1. Polytopes cut out by roots

3.2. Polytopes spanned by roots

3.3. Other Graph Polytopes

3.4. Lecture hall polytopes

3.5. Smooth Polytopes

3.6. The Gröbner fan and the toric Hilbert scheme

Chapter 4. Dilations and the KMW Theorem

4.1. KMW numbers in dimension three

4.2. Canonical triangulation of a dilated simplex

4.3. Reducing the volume of simplices in the dilation

4.4. A proof of the KMW Theorem

4.5. An effective version of the KMW-Theorem

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470465308

1470465302

OCLC:

1259594323