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Abstract: We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field . More precisely, given and a unimodular vector (that is, such that ), the algorithm computes a matrix M in whose determinant is a monomial such that , and thus is a completion of to an invertible matrix. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to decipher constructively a lemma of Suslin which played a central role in his second solution of Serre’s problem on projective modules over polynomial rings. This lemma says that for a commutative ring if where is monic and , then there exist such that, denoting by the first coordinate of , we have . By the constructive proof we give, Suslin’s proof of Serre’s problem becomes fully constructive. Moreover, the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1. [Copyright &y& Elsevier]