Coprime Ehrhart Theory and Counting Free Segments

Abstract A lattice polytope is free (or empty) if its vertices are the only points it contains. In context of valuation theory, Klain [16] proposed to study functions $\alpha _i(P;n)$ that count number polytopes in $nP$ with $i$ vertices. For $i=1$, this famous Ehrhart polynomial; for $i> 3$, computation likely impossible; and $i=2,3$ computationally challenging. paper, we develop a theory coprime relatively prime coordinates use compute _2(P;n)$ unimodular simplices. We show function can be explicitly determined from polynomial give some applications combinatorial counting.