An algorithm to compute the set of characteristics of a system of polynomial equations over the integers

We describe a (finite) algorithm to determine the set of characteristics of a system of polynomial equations with integer coefficients by using the theory of Gröbner bases. This gives us a proof that the set of characteristics must be either finite and not containing zero, or containing zero and co-finite. Another, algebraic, proof of this is given in the appendix. These results carry over to systems of polynomial equations over a principal ideal domain and also yields an algorithm for finding the characteristic set of a matroid.