17 - The CoCoA 3 Framework for a Family of Buchberger-like Algorithms

Summary

Abstract

In this paper we report how we have implemented in release 3.3 of the system CoCoA algorithms for computing Gröbner bases, syzygies, and minimal free resolution of submodules of finitely generated free modules over polynomial rings. In particular we describe the environment we have built where these algorithms are obtained as special instances of a single parametrized algorithm. The outcome is a framework which offers not only efficiency in computing, but also efficiency in designing new algorithms; moreover, it allows the user to interactively execute the above computations, and easily customize the way they are carried out.

Introduction

Computing Gröbner bases, syzygies, and finite free resolutions of ideals and of submodules of finitely generated free modules over polynomial rings is a fundamental task in Computational Commutative Algebra. Nowadays there are several specialized computer algebra systems, developed by researchers in the field, which efficiently perform such computations, for instance Asir (Noro and Takeshima 1992), CoCoA (Capani et al. 1996), Macaulay (Bayer and Stillman 1992), Macaulay 2 (Grayson and Stillman 1996), Singular (Greuel et al. 1996).

The cornerstone in this area has been the (now) classical algorithm due to Buchberger (1965) which produces a Gröbner basis of an ideal in a polynomial ring R over a field, starting from any of its possible finite sets of generators. This algorithm can be suitably modified (Schreyer 1980, Spear 1977) in such a way that it also computes the syzygy module of a list of polynomials.