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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: We give an algorithm for the well-known result asserting that if R is a polynomial ring in a finite number of variables over a Noetherian ring A of Krull dimension , then for , acts transitively on . For technical reasons we demand that the Noetherian ring A has a theory of Gröbner bases and contains an infinite set such that for each . The most important guiding examples are affine rings and localizations of polynomial rings , with K an infinite field. Moreover, we give an algorithmic proof of Suslin''s stability theorem over these rings. For the purpose to prepare the ground for this algorithmic generalizations of the Quillen–Suslin theorem (corresponding to the particular case A is a field), we will give in the first section a constructive proof of an important lemma of Suslin which is the only nonconstructive step in Suslin''s second elementary solution of Serre''s conjecture. This lemma says that for a commutative ring A, if where is monic and , then there exist such that . Thanks to this constructive proof, Suslin''s second proof of Serre''s conjecture becomes fully constructive. [Copyright &y& Elsevier]

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]