A heuristic selection strategy for lexicographic Gröner bases?

Abstract

ABSTMCT It is well known that the computation of lexicographic Grobner bases using Buchberger's algorithm is more difficult than the computation of Gr6bner bases with respect to totat degree orderings. The lexicographic algorithm is particularly susceptible to the problem of intermediate expression swell; that is, intermediate polynomials may be far larger than those which make up the final basis. To some extent, this is a function of "selection strategy", i.e. the order in which S-polynomials are used to extend a partial basis. We argue and provide empirical evidence thatj_or thelexicographic ordering (in direct contrast to the case of degree orderings), a simple heuristic strategy will in practice control intermediate growth more effectively than the normal strategy based on the lexicographic term ordering alone. The result is usually a much more efficient computation, even for ideals of nonzero dimension (where lexicographic basesare uniquely well suited to solving the corresponding algebraic systemsof equations).

Fast hierarchical clustering and other applications of dynamic closest pairs

“…see [Adams and Loustaunau 1994, p. 130]): select f i and f j for which the least common multiple of the leading terms of f i and f j is as small as possible in the term ordering. (Other selection strategies have also been proposed [Czapor 1991;Giovini et al 1991] and it seems likely that our methods apply as well to them. )…”

Optimal Ladder Filters

“…The problem with this method is that its complexity in terms of computer memory and computation time explosively rises with the order of the filter. Much research has been done to reduce this complexity [12]- [15].…”