Ehrhart polynomial for lattice squares, cubes and hypercubes

TL;DR

This paper investigates Ehrhart polynomials for lattice squares, cubes, and hypercubes in dimensions 2 to 4, providing complete descriptions for squares and cubes, and conjecturing formulas for hypercubes.

Contribution

It offers a complete characterization of Ehrhart polynomials for lattice squares and cubes, and proposes a conjecture for hypercubes based on computed coefficients.

Findings

Complete Ehrhart polynomial descriptions for squares and cubes.

Computed one coefficient for hypercubes and identified linear relations.

Formulated a conjecture for hypercube Ehrhart polynomials.

Abstract

In this paper we are constructing integer lattice squares, cubes or hypercubes in $\mathbb R^d$ with $d\in \{2,3,4\}$. For squares and cubes we find a complete description of their Ehrhart polynomial. For hypercubes, we compute one of the coefficients and show that there exists a simple linear relation between the other two unknown coefficients. This allows as to formulate a conjecture of what the Ehrhart polynomial is in this case.