A Parallel Implementation of Buchberger's Algorithm overZpforp ≤ 31991

Gröbner bases of ideals of polynomials are known to have many applications. They have been applied to problems in commutative algebra, statistics, graph theory, robotics and differential equations. Their use as a research tool, however, is limited by their computational complexity. These two facts have inspired numerous attempts to parallelize Buchberger's algorithm to compute them. In this paper, we describe a parallel implementation developed on the Cray T3D using the extensions to C provided by ac. The program is based on the publicly available package Macaulay which computes Gröbner bases of homogeneous ideals overZpfor primesp≤ 31991. The efficiency is nearly 100% on up to 16 processors for moderately sized problems. Above 16 processors, the efficiency drops.