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Abstract: A well-known lemma of Suslin says that for a commutative ring if is unimodular where is monic and , then there exist such that the ideal generated by equals . This lemma played a central role in the resolution of Serre’s Conjecture. In the case where contains a set of cardinality greater than such that is invertible for each in , we prove that the can simply correspond to the elementary operations , , where . These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in to using elementary operations in the case where is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds. [Copyright &y& Elsevier]

Recently, Lombardi, Quitté and Yengui have given a Gröbner-free algorithm which computes the V -saturation of any finitely generated submodule of V [ X ] n , where V is a valuation domain. The goal of this paper is to clarify this algorithm, to give precise complexity bounds, and a complete submodule membership test for the saturation. As application, we give precise degree bounds on syzygies over V [ X ] . [ABSTRACT FROM AUTHOR]