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Integer decomposition property of polytopes

Sharon Robins

Abstract

We study the integer decomposition property of lattice polytopes associated with the $n$-dimensional smooth complete fans with at most $n+3$ rays. Using the classification of smooth complete fans by Kleinschmidt and Batyrev and a reduction to lower dimensional polytopes, we prove the integer decomposition property for lattice polytopes in this setting.

Integer decomposition property of polytopes

Abstract

We study the integer decomposition property of lattice polytopes associated with the -dimensional smooth complete fans with at most rays. Using the classification of smooth complete fans by Kleinschmidt and Batyrev and a reduction to lower dimensional polytopes, we prove the integer decomposition property for lattice polytopes in this setting.

Paper Structure

This paper contains 3 sections, 5 theorems, 77 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result

Key Result

Theorem 1.4

If $P,Q$ are two lattice polytopes whose coarsest common refinement of normal fans has at most $n+3$ rays in $\mathbb{R}^n$, then the pair $(P,Q)$ has the IDP.

Figures (8)

  • Figure 1: An example of polytopes that do not have IDP
  • Figure 2: Refining the normal fan into a smooth fan
  • Figure 3:
  • Figure 4: Projection of a polytope
  • Figure 5: Polytopes $P^{\Delta},Q^{\Delta},(P+Q)^{\Delta}$
  • ...and 3 more figures

Theorems & Definitions (14)

  • Conjecture 1.1: Oda’s Conjecture
  • Theorem 1.4
  • Conjecture 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: batyrev1991
  • Example 2.5
  • Theorem 2.6: Kleinschmidt1988
  • Theorem 2.7: MR2551605
  • Theorem 2.8: batyrev1991
  • ...and 4 more