Hermite normal forms and δ-vectors

Let δ ( P ) = ( δ 0 , δ 1 , … , δ d ) be the δ-vector of an integral polytope P ⊂ R N of dimension d. Following previous work on the characterization of δ-vectors with ∑ i = 0 d δ i ⩽ 3 , all the possible δ-vectors with ∑ i = 0 d δ i = 4 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 ( δ 0 , δ 1 , … , δ d ) , where ∑ i = 0 d δ i ⩽ 4 .