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It is well-known that for any commutative unitary ring $\mathbf{R}$, the Serre conjecture ring $\mathbf{R}\langle X \rangle$, i.e., the localization of the univariate polynomial ring $\mathbf{R}[X]$ at monic polynomials, is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. Consequently, defining by induction $\mathbf{R}\langle X_1,\ldots,X_n \rangle:=(\mathbf{R}\langle X_1,\ldots,X_{n-1}\rangle)\langle X_n\rangle$, the ring $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. The fact that $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain when $\mathbf{R}$ is a valuation domain of Krull dimension $\leq 1$ was the cornerstone of Brewer and Costa's theorem stating that if $\mathbf{R}$ is a one-dimensional arithmetical ring then finitely generated projective $\mathbf{R}[X_1,\dots,X_n]$-modules are extended. It is also the key of the proof of the Gr\"obner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain $\mathbf{R}$ of Krull dimension $\leq 1$, any $n \in \mathbb{N}_{>0}$, and any finitely generated ideal $I$ of $\mathbf{R}[X_1, \dots, X_n]$, the ideal $\operatorname{LT}(I)$ generated by the leading terms of the elements of $I$ with respect to the lexicographic monomial order is finitely generated. Since the ring $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ can also be defined directly as the localization of the multivariate polynomial ring $\mathbf{R}[X_1,\dots,X_n]$ at polynomials whose leading coefficients according to the lexicographic monomial order with $X_1, Comment: 12 pages, no figure

International audience

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 B��zout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

International audience; We give an algorithm for computing the $V$-saturation of any finitely generated submodule of $V[X_1,... ,X_k]^m (k \in N, m \in N^*)$, 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 $V[X_1,... ,X_k]$.

International audience

International audience

International audience

In this paper, I present a new decision procedure for the ideal membership problem for polynomial rings over principal domains using discrete valuation domains. As a particular case, I solve 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 $mathbb{Z}[X_1,ldots, X_n]$. The techniques utilized are easily generalizable to Dedekind domains. In order to avoid the expensive complete factorization in the basic principal ring, I introduce the notion of ``dynamical Gr"obner bases" of polynomial ideals over a principal domain. As application, I give an alternative dynamical solution to ``Kronecker's problem".

We constructively prove that for any ring R with Krull dimension ⩽d, the ring R〈X〉 locally behaves like the ring R(X) or a localization of a polynomial ring of type (S−1R)[X] with S a multiplicative subset of R such that the Krull dimension of S−1R is ⩽d−1. 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.