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 Z[X 1 , . . . , X n ]. The notions of Gröbner bases over Noetherian valuation rings and dynamical Gröbner bases over principal rings have applications in error correcting codes.

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We give a constructive deciphering for a generalization of the Quillen-Suslin theorem due to Maroscia and Brewer & Costa stating that finitely generated projective modules over R[X 1 , . . . , X n ], where R is a Prüfer domain with Krull dimension ≤ 1, are extended from R.In this paper all rings are commutative and unitary. We follow the philosophy developed in the papers [2][3][4][5][11][12][13][14][15][16][17][18]23,24]. The main goal is to find the constructive content hidden in the abstract proofs of concrete theorems.The general method consists in replacing some abstract ideal objects whose existence is based on the principle of the excluded middle and the axiom of choice by incomplete specifications of these objects. This paper is a sequel to [18]. We continue to develop the constructive rereading of abstract methods that use local-global principles. Our explicit proofs are obtained by a deciphering of the arguments contained in the original abstract proofs. We think that this is a first step in the achievement of Hilbert's program for abstract algebra methods.

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