Title

Ehrhart polynomial for lattice squares, cubes and hypercubes

Document Type

Article

Publication Date

1-1-2019

Publication Title

Revue Roumaine de Mathematiques Pures et Appliquees

Volume

64

First Page

57

Last Page

80

Keywords

Ehrhart polynomial, Icube, Lattice cube, Lattice square, Linear Diophantine equations, Pythagorean quadruple, Quaternions, Twin vectors

Abstract

© 2019 Editura Academiei Romane. All rights reserved. In this paper, we are constructing integer lattice squares, cubes or hypercubes in R n with n ∈ {2, 3, 4}. We nd a complete description of their Ehrhart polynomial. We characterize all the integer squares in R 4 , in terms of two Pythagorean quadruple representations of the form a 2 + b 2 + c 2 = d 2 , and then prove a parametrization in terms of two quaternions of all such squares. We introduce the sequence of almost perfect squares in dimension n. In dimension two, this is very close to the sequence A194154 (in OEIS).

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