Criteria for unimodular triangulations and the integer decomposition property

Determine effective criteria on a vector a in N^d that ensure the associated one-row Hermite normal form simplex S_a admits a unimodular triangulation or satisfies the integer decomposition property.

Background

The paper completely characterizes the relevant properties for the special one-row family a=(N-1,...,N-1,N), establishing that a unimodular triangulation and the integer decomposition property occur exactly when N has the form kd or kd+1. It also proves failure of the integer decomposition property for the other values of N in this family.

The authors emphasize that determining whether a lattice polytope has the integer decomposition property is generally difficult, even for restricted families. They therefore ask for effective criteria applicable to arbitrary vectors a in N^d and to the associated one-row Hermite normal form simplices.