Ehrhart-Equivalence, Equidecomposability, and Unimodular Equivalence of Integral Polytopes

TL;DR

This paper explores the relationships between Ehrhart-equivalence, equidecomposability, and unimodular equivalence of integral polytopes, proposing conjectures, algorithms, and new theoretical insights into their classification.

Contribution

It establishes connections between different equivalence notions of integral polytopes, introduces an algorithm for unimodular equivalence, and proves the existence of non-equivalent simplices in all dimensions.

Findings

Proposes a conjecture relating Ehrhart-equivalence and equidecomposability.

Develops an algorithm to determine unimodular equivalence of simplices.

Proves the existence of non-unimodular equivalent simplices in all dimensions.

Abstract

Ehrhart polynomials are extensively-studied structures that interpolate the discrete volume of the dilations of integral $n$-polytopes. The coefficients of Ehrhart polynomials, however, are still not fully understood, and it is not known when two polytopes have equivalent Ehrhart polynomials. In this paper, we establish a relationship between Ehrhart-equivalence and other forms of equivalence: the $\operatorname{GL}_n(\mathbb{Z})$-equidecomposability and unimodular equivalence of two integral $n$-polytopes in $\mathbb{R}^n$. We conjecture that any two Ehrhart-equivalent integral $n$-polytopes $P,Q\subset\mathbb{R}^n$ are $\operatorname{GL}_n(\mathbb{Z})$-equidecomposable into $\frac{1}{(n-1)!}$-th unimodular simplices, thereby generalizing the known cases of $n=1, 2, 3$. We also create an algorithm to check for unimodular equivalence of any two integral $n$-simplices in $\mathbb{R}^n$. We then find and prove a new one-to-one correspondence between unimodular equivalence of integral $2$-simplices and the unimodular equivalence of their $n$-dimensional pyramids. Finally, we prove the existence of integral $n$-simplices in $\mathbb{R}^n$ that are not unimodularly equivalent for all $n \ge 2$.