Efficiently Computing Minimal Sets of Critical Pairs

In the computation of a Gr"obner basis using Buchberger’s algorithm, a key issue for improving the efficiency is to produce techniques for avoiding as many unnecessary critical pairs as possible. A good solution would be to avoid all non-minimal critical pairs, and hence to process only a_minimal_ set of generators of the module generated by the critical syzygies. In this paper we show how to obtain that desired solution in the homogeneous case while retaining the same efficiency as with the classical implementation. As a consequence, we get a new Optimized Buchberger Algorithm.