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It is well-known that for any integral domain R , the Serre conjecture ring R 〈 X 〉 , i.e., the localization of the univariate polynomial ring R [ X ] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R 〈 X 1 , ... , X n 〉 : = (R 〈 X 1 , ... , X n − 1 〉) 〈 X n 〉 , the ring R 〈 X 1 , ... , X n 〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R 〈 X 1 , ... , X n 〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R [ X 1 , ... , X n ] -modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n ∈ N > 0 , and any finitely generated ideal I of R [ X 1 , ... , X n ] , the ideal 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 R 〈 X 1 , ... , X n 〉 can also be defined directly as the localization of the multivariate polynomial ring R [ X 1 , ... , X n ] at polynomials whose leading coefficients according to the lexicographic monomial order with X 1 < X 2 < ⋯ < X n is 1, we propose to generalize the fact that R 〈 X 1 , ... , X n 〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case. [ABSTRACT FROM AUTHOR]

We prove that if a valuation domain V has Krull dimension ≤ 1 then for every finitely generated ideal I of V [ X ] the trailing terms ideal t t (I) of I (that is, the ideal generated by the trailing terms of the nonzero polynomials in I) is also finitely generated. Heinzer and Papick (1989) and then Lombardi, Schuster and Yengui (2012) have shown the analogous result for leading terms ideals. The proof we give is both simple and constructive. The same result is valid for Prüfer domains. [ABSTRACT FROM AUTHOR]

In this paper, we address the following question: for a nonzero finitely generated ideal I of a multivariate polynomial ring R [ X 1 , ... , X n ] over a coherent ring R , fixing a monomial order < on R [ X 1 , ... , X n ] , is the trailing terms ideal tt (I) of I (that is, the ideal generated by the trailing terms of the nonzero polynomials in I) finitely generated? We show that while tt (I) can be nonfinitely generated, it is always countably generated when the monomial order is Noetherian (graded monomial orders as instances). [ABSTRACT FROM AUTHOR]

Abstract: We construct an example of a finitely generated ideal I of , where V is a one-dimensional valuation ring, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if V is a valuation ring with Krull dimension ⩽1, then for any finitely generated ideal I of , the leading terms ideal of I is also finitely generated. The valuation rings satisfying this latter property will be called 1-Gröbner and are studied in this paper. [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]