Deterministically Computing Reduction Numbers of Polynomial Ideals

Abstract

We discuss the problem of determining reduction numbers of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computations in a polynomial ring with (n − dim I) dim I parameters and n − dim I variables. The second one computes via a Gröbner system the set of all reduction numbers of the ideal I and thus in particular also its big reduction number. However, it requires computations in a ring with n dim I parameters and n variables.