Ehrhart Polynomials of Generic Orthotopes

TL;DR

This paper develops a theory of Ehrhart polynomials for integral generic orthotopes, revealing a relation between lattice points and floral types via local polynomials linked to read-once Boolean functions.

Contribution

It introduces a novel framework connecting Ehrhart polynomials of orthotopes with read-once Boolean functions and floral types, expanding geometric combinatorics.

Findings

Established a formula relating lattice points to floral types in orthotopes

Introduced local polynomials for read-once Boolean functions

Connected Ehrhart theory with Boolean function representations

Abstract

A generic orthotope is an orthogonal polytope whose tangent cones are described by read-once Boolean functions. The purpose of this note is to develop a theory ofEhrhart polynomials for integral generic orthotopes. The most remarkable part of this theory is a relation between the number of lattice points in an integral generic orthotope $P$ and the number of unit cubes in $P$ of various floral types. This formula is facilitated through the introduction of a set of "local polynomials" defined for every read-once Boolean function.