Generating Prime Ideals in the Minkowski Ring of Poly top es

The Minkowski ring of a set of points V ₁,…,V _n in Euclidean space, is an integral domain that is isomorphic to a factor ring of Z[X ₁,…,X _n ]. We give both upper and lower bounds on the number of generators needed for the prime ideal of relations in terms of certain rational relations on the set V ₁,…,V _n . Additionally we show that these bounds are tight. Since the Minkowski ring on a set of points is isomorphic to a semigroup ring, our results extend some of the work of Herzog [H] and Bresinsky [B].