Coprime Ehrhart theory and counting free segments

TL;DR

This paper introduces coprime Ehrhart functions to analyze free lattice polytopes, providing explicit formulas for counting free segments in unimodular simplices and linking these to classical Ehrhart polynomials.

Contribution

It develops a new theory of coprime Ehrhart functions and applies it to compute the number of free segments in unimodular simplices, connecting it to existing Ehrhart theory.

Findings

Coprime Ehrhart functions can be explicitly derived from Ehrhart polynomials.

The theory enables counting free segments in unimodular simplices.

Applications to combinatorial counting are demonstrated.

Abstract

A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $\alpha_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $\alpha_2(P;n)$ for unimodular simplices. We show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and we give some applications to combinatorial counting.