Existence of regular unimodular triangulations of dilated empty simplices

Given integers k and m with k≥2 and m≥2, let P be an empty simplex of dimension (2k−1) whose δ-polynomial is of the form 1+(m−1)tk. In the present paper, the necessary and sufficient condition for the kth dilation kP of P to have a regular unimodular triangulation will be presented.