When does 〈T〉〈T〉 equal sat(T)?

Given a regular chain TT, we aim at finding an efficient way for computing a system of generators of sat(T), the saturated ideal of TT. A natural idea is to test whether the equality 〈T〉=sat(T) holds, that is, whether TT generates its saturated ideal. By generalizing the notion of primitivity from univariate polynomials to regular chains, we establish a necessary and sufficient condition, together with a Gröbner basis free algorithm, for testing this equality. Our experimental results illustrate the efficiency of this approach in practice.

► Let TT be a regular chain in a multivariate polynomial ring over a field. ► A necessary and sufficient condition for TT to generate its saturated ideal is given. ► In addition, we establish an algorithm for testing this condition. ► This provides us with an effective criterion for saturated ideal inclusion. ► Experimental results are reported.