Integer decomposition property of dilated polytopes

TL;DR

This paper investigates the integer decomposition property of dilated convex polytopes, proposing combinatorial invariants to understand when these dilations possess this property.

Contribution

It introduces and studies combinatorial invariants related to the integer decomposition property of dilated polytopes, advancing understanding of their structural characteristics.

Findings

Proposed combinatorial invariants for dilated polytopes.

Characterized conditions for integer decomposition property.

Analyzed the relationship between invariants and polytope dilations.

Abstract

Let $\mathcal{P} \subset \mathbb{R}^N$ be an integral convex polytope of dimension $d$ and write $k \mathcal{P}$, where $k = 1, 2, \ldots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k = 1, 2, \ldots$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^N$, there exist $\alpha_{1}, \ldots, \alpha_k$ belonging to $\mathcal{P} \cap \mathbb{Z}^N$ such that $\alpha = \alpha_1 + \cdots + \alpha_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.