On an implementation of standard bases and syzygies in SINGULAR
In a previous article [Arch. Math. 66, No. 2, 163-176 (1996; see the preceding review)], the second and fifth authors described a generalization of the algorithms of Buchberger and Mora to the case of any ordering on the monomials in a polynomial ring that is compatible with the semigroup structure. (The variables may have negative, positive, or zero weights.) In the present article, the authors describe the implementation of these ideas in the system SINGULAR. In the first part of the paper, the implementation of the standard basis algorithm is presented. Two methods that can improve the efficiency of this algorithm are the HCtest and ecartMethod. The HCtest is applicable only to 0-dimensional ideals and involves computing the minimal monomial not contained in the initial ideal (the highest corner) and discarding all bigger monomials in further computations. The ecartMethod is applicable in all cases and provides an automatic choice of a weight vector of positive integers such that the weight ecart of the input polynomials become as small as possible. Results of twenty standard basis computations with various options (e.g. Lazard's method, Mora's algorithm with sugar, HCtest, ecartMethod) are given. In the last part of the paper, the authors compare different methods for computing syzygies and minimal resolutions in twenty cases. Included in the first implementation of F.-O. Schreyer's algorithm for computing syzygies. No single strategy is best in all cases.
scientific article; zbMATH DE number 878206
scientific article