Algorithm 628

Algorithm 628. An algorithm for constructing canonical bases of polynomial ideals. The paper is mainly a report concerning the FORTRAN implementation of an algorithm for constructing (canonical) Gröbner bases (GB) for polynomial ideals. Roughly speaking, the problem is: ”Given a finite set F of polynomials in K[x 1 ,···,x n ], find a finite set G of polynomials in K[x 1 ,···,x n ] such that ideal (F)=ideal (G) and G is a Gröbner basis.” A GB is characterized by the property that some reduction relation → F has the Church-Rosser property [see B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, SIGSAM Bull. 10, No.3, 19-29 (1976)]. Some examples of constructing specific GB using the algorithm are presented, in order to indicate the usefulness of these bases for constructive polynomial ideal theory and computer algebra (many decision and computation problems for polynomial ideals may be solved easily for ideals given by a GB than for an arbitrary basis).