Hilbert Functions and the Buchberger Algorithm

In this paper we show how to use the knowledge of the Hilbert–Poincaré series of an idealIto speed up the Buchberger algorithm for the computation of a Gröbner basis. The algorithm is useful in the change of ordering and in the validation of modular computations, also with tangent cone orderings; speeds the direct computation of a Gröbner basis if the ideal is a complete intersection, e.g. in the computation of cartesian from parametric equations, can validate or disprove a conjecture that an ideal is a complete intersection, and is marginally useful also when the conjecture is false. A large set of experiments is reported.