Hermite normal forms and δ-vectors

Let δ(P)=(δ0,δ1,…,δd) be the δ-vector of an integral polytope P⊂RN of dimension d. Following previous work on the characterization of δ-vectors with , all the possible δ-vectors with are classified by means of simplices. We obtain our results by considering—by means of Hermite normal forms of square matrices—the classification of integral simplices with a given δ-vector (δ0,δ1,…,δd), where .