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We provide constructive versions of Hilbert's syzygy theorem for Z and Z/NZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated. [ABSTRACT FROM AUTHOR]

We propose a new version of Noether normalization theorem over Bezout domains of Krull dimension one (the ring Z of integers as main example). We also show that one can avoid branching when computing dynamical Gröbner bases over a Bezout domain of Krull dimension one. [ABSTRACT FROM AUTHOR]

Abstract: In this corrigendum, I correct a mild error in a technical lemma of Yengui (2006) . This error affected some results and examples in Hadj Kacem and Yengui (2010) and Yengui (2006) . [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]