Coprime Ehrhart theory and counting free segments

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 $α_i(P;n)$ 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 $α_2(P;n)$ 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.

No reviews yet.

No comments yet.