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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: In this paper, we introduce the notion of “dynamical Gröbner bases” of polynomial ideals over a principal ring. As application, we solve dynamically a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker''s problem,” that is the problem of finding a decision procedure for the ideal membership problem for . The notions of Gröbner bases over Noetherian valuation rings and dynamical Gröbner bases over principal rings have applications in error correcting codes. [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]

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]

We give an algorithm for computing the V -saturation of any finitely-generated submodule of V [ X 1 , … , X k ] m ( k ∈ N , m ∈ N ⁎ ), where V is a valuation domain. Our algorithm is based on a notion of “echelon form” which ensures its correctness. The proposed algorithm terminates when two (Hilbert) series on the quotient field and the residue field of V coincide. As application, our algorithm computes syzygies over V [ X 1 , … , X k ] . [ABSTRACT FROM AUTHOR]

Abstract: We constructively prove that for any ring R with Krull dimension ⩽d, the ring locally behaves like the ring or a localization of a polynomial ring of type with S a multiplicative subset of R such that the Krull dimension of is . As an application, we give a simple and constructive proof of the Lequain–Simis Induction Theorem which is an important variation of the Quillen Induction Theorem. [Copyright &y& Elsevier]