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

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 $n \ge 2$.