IDP for 2-Partition Maximal Symmetric Polytopes

TL;DR

This paper introduces a framework to analyze the integer decomposition property of symmetric polytopes, demonstrating its application to a specific class called 2-partition maximal polytopes in three-dimensional space.

Contribution

It presents a novel approach for proving the integer decomposition property for symmetric polytopes, focusing on a special case in three dimensions.

Findings

Established the integer decomposition property for 2-partition maximal polytopes in 3

Developed a method involving saturated Newton polytope of certain polynomials

Provided a framework applicable to symmetric polytopes in geometric combinatorics

Abstract

We provide a framework for which one can approach showing the integer decomposition property for symmetric polytopes. We utilize this framework to prove a special case which we refer to as $2$-partition maximal polytopes in the case where it lies in a hyperplane of $\mathbb{R}^3$. Our method involves proving a special collection of polynomials have saturated Newton polytope.