“One sugar cube, please” or selection strategies in the Buchberger algorithm

In this paper redescribe some experimentti findings on selection strategies for Gr6bner basis computation with the Buchberger algorithm. In particular, the results suggest that the “sugar flavor” of the “normal selection”, implemented first in COCOA, then in AlPI, [14], [15] (up to now in the muLISP version, in a short time in the COMMON-LISP version, including the parallel version, [1]) and now in SCRATCHPAD-II, is the best choice for a selection strategy. It has to be combined with the “straightforward” simplification strategy and with a special form of the Gebauer-Moller criteria to obtain the best results. The idea of the “sugar flavor” is the following: the Buchberger algorithm for homogeneous ideals, with degreecompatible term ordering and normal selection strategy, usually works fine. Homogenizing the basis of the ideal is good for the strategy, but bad for the basis to be computed. The sugar flavor computes, for every polynomial in the course of the algorithm, ‘(the degree that it would have if computed with the homogeneous algorithm”, and uses this phantom degree (the sugar) only for the selection strategy. We have tested several examples with different selection strategies, and the sugar flavor has proved to be always the best choice or very near to it. The comparison between the different variants of the sugar flavor has been made, but the results are up to now inconclusive. We include a complete deterministic description of the Buchberger algorithm as it was used in our experiments.l