For computing free resolutions over a polynomial ring the usual approach consists in iterating the Buchburger's algorithm for each module in the resolution. In this paper, we propose one single algorithm which can be viewed as a generalization of Buchberger's to chain complexes. The algorithm is based on the use of syzygies, due to Mo¨ller, Mora and Traverso, as criteria for avoiding useless computation of S-polynomials. Some strategies for the pairs selection in complexes are studied and tested in some experiments.
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