On implementing Buchberger's algorithm for Grobner bases

An implementation in the Maple system of Buchberger's algorithm for computing Gröbner bases is described. The efficiency of the algorithm is significantly affected by choices of polynomial representations, by the use of criteria, and by the type of coefficient arithmetic used for polynomial reductions. The improvement possible through a slightly modified application of the criteria is demonstrated by presenting time and space statistics for some sample problems. A fraction-free method for polynomial reduction is presented. Timings on problems with integer and polynomial coefficients show that a fraction-free approach is recommended.