Search

We propose a counterexample to the Grobner ring conjecture in the irrational case. More precisely, we construct a valuation domain V of Krull dimension 1 and a finitely generated ideal J of V [ X , Y ] equipped with some irrational monomial order LT ( J ) generated by the leading terms of the elements of J is not finitely generated.

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 Grobner bases over a Bezout domain of Krull dimension one.

Recently, Lombardi, Quitte and Yengui have given a Grobner-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 .

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 ] .

We construct an example of a finitely generated ideal I of R [ X ] , where R is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if R is a domain with Krull dimension ≤1, then for any finitely generated ideal I of R [ X ] , the leading terms ideal of I is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain R and any a , b ∈ R , the ideal of R [ X ] generated by the leading terms of 〈 1 + a X , b 〉 is finitely generated.

We construct an example of a finitely generated ideal I of V [ X ] , 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 V [ X ] , the leading terms ideal of I is also finitely generated. The valuation rings satisfying this latter property will be called 1-Grobner and are studied in this paper.

In this paper, we extend the notion of “dynamical Grobner bases” introduced by the second author to Dedekind rings (with zero divisors). As an application, we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in Dedekind rings. We also give a partial positive answer to a conjecture about Grobner rings.

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring R = K [ X 1 ± , … , X k ± ] over an infinite field K . More precisely, given n ⩾ 3 and a unimodular vector V = t ( v 1 , … , v n ) ∈ R n (that is, such that 〈 v 1 , … , v n 〉 = R ), the algorithm computes a matrix M in M n ( R ) whose determinant is a monomial such that M V = t ( 1 , 0 , … , 0 ) , and thus M - 1 is a completion of V to an invertible matrix.

We prove constructively that for any ring R of Krull dimension ⩽1 and n ⩾ 3 , the group E n ( R [ X ] ) acts transitively on Um n ( R [ X ] ) . In particular, we obtain that for any ring R with Krull dimension ⩽1, all finitely generated stably free modules over R [ X ] are free. This settles the long-standing Hermite ring conjecture for rings of Krull dimension ⩽1.

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.

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[X1,…,Xn]. The notions of Gröbner bases over Noetherian valuation rings and dynamical Gröbner bases over principal rings have applications in error correcting codes.

A well-known lemma of Suslin says that for a commutative ring A if (v"1(X),...,v"n(X))@?(A[X])^n is unimodular where v"1 is monic and n>=3, then there exist @c"1,...,@c"@?@?E"n"-"1(A[X]) such that the ideal generated by Res(v"1,e"1.@c"1^t(v"2,...,v"n)),...,Res(v"1,e"1.@c"@?^t(v"2,...,v"n)) equals A. This lemma played a central role in the resolution of Serre's Conjecture. In the case where A contains a set E of cardinality greater than degv"1+1 such that y-y^' is invertible for each y y^' in E, we prove that the @c"i can simply correspond to the elementary operations L"1->L"1+y"i@?"j"="2^n^-^1u"j"+"1L"j, 1@?i@?@?=degv"1+1, where u"1v"1+...+u"nv"n=1. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in K[X"1,...,X"k] to ^t(1,0,...,0) using elementary operations in the case where K is an infinite field. Another feature of this paper is that it shows that the concrete local-global principles can produce competitive complexity bounds.

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X1,…,Xn〉 residually Jaffard (resp., totally Jaffard) when R(X1,…,Xn) is ? We construct a three-dimensional local ring R such that R(X1,…,Xn) is totally Jaffard (and hence, residually Jaffard) whereas R〈X1,…,Xn〉 is not residually Jaffard (and hence, not totally Jaffard).

We construct two counterexamples to the open questions : is R 〈 n 〉 strong S (resp. catenary) when R ( n ) is ? The first example is a ring R such that R ( n ) is strong S and R 〈 n 〉 is not. The second is a stably strong S -domain R such that for all n ≥1 and n =∞, R ( n ) is catenary and R 〈 n 〉 is not.

We propose to give an answer to two open questions: is R〈X〉 strong S (resp., catenarian) when R(X) is? We construct a ring R such that R(X) is both catenarian and strong S whereas R〈X〉 is neither catenarian nor strong S.