Dynamical Gröbner bases over Dedekind rings
type: Item
instance of: scholarly article
title: Dynamical Gröbner bases over Dedekind rings
zbMATH Open document ID: 1200.13047
author: Amina Hadj Kacem
author: Ihsen Yengui
published in: Journal of Algebra
publication date: 2010-07-20T00:00:00Z
review text: The notion of dynamical Gröbner bases for ideals in \(R[x_1, \ldots, x_n]\), \(R\) a Dedekind ring (with zero divisors) is introduced: Let \(>\) be a monomial ordering on \(R[x_1, \ldots, x_n]\) and \(I\subset R[x_1, \ldots, x_n]\) an ideal. \(G=\{g_1, \ldots, g_l\}\subset I\) is called a special Gröbner basis for \(I\) if \(I=\langle g_1, \ldots, g_l\rangle\) and the set \(\{\text{LC}(g_1), \ldots, \text{LC}(g_l)\}\) is totally ordered under division and for each \(i\neq j\) \( \overline{S(g_i, g_j)}^G=0\). A set \(\{(S_1, G_1), \ldots, (S_k, G_k)\}\) is called a dynamical Gröbner basis for \(I\) if \(S_1, \ldots, S_k\) are finitely generated comaximal multiplicative subsets of \(R\) and in each localization \((S_i^{-1} R)[x_1, \ldots, x_n] \) \(G_i\) is a special Gröbner basis for \(I\). As an application the ideal membership problem and the computation of syzygies are treated.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 5758104
zbMATH Keywords: Gröbner basis
zbMATH Keywords: dynamical Gröbner basis
zbMATH Keywords: ideal membership problem
zbMATH Keywords: principal rings
zbMATH Keywords: Dedekind rings
zbMATH Keywords: Gröbner rings
zbMATH Keywords: constructive mathematics
describes a project that uses: SINGULAR
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.04.014
OpenAlex ID: W2018462023
cites work: Q4305609
cites work: Q4375594
cites work: Factoring into coprimes in essentially linear time
cites work: Q3336682
cites work: Approximating rings of integers in number fields
cites work: Hidden constructions in abstract algebra. Krull Dimension of distributive lattices and commutative rings
cites work: Dynamical method in algebra: Effective Nullstellensätze
cites work: Algorithmic theory of arithmetic rings, Prüfer rings and Dedekind rings. (Théorie algorithmique des anneaux arithmétiques, des anneaux de Prüfer et des anneaux de Dedekind.)
cites work: Sketches and computation – II: dynamic evaluation and applications
cites work: A constructive comparison of the rings \(R(X)\) and \(R\langle X\rangle\) and application to the Lequain-Simis induction theorem
cites work: Q2781760
cites work: Krull dimension, Nullstellensätze and dynamical evaluation
cites work: Hidden constructions in abstract algebra. VI: The theorem of Maroscia and Brewer {\&} Costa
cites work: A course in constructive algebra
cites work: On the construction of Gröbner bases using syzygies
cites work: Q4413922
cites work: Gröbner bases with coefficients in rings
cites work: Nontrivial Uses of Trivial Rings
cites work: Syzygies of canonical curves and special linear series
cites work: A standard basis approach to syzygies of canonical curves.
cites work: Q3856855
cites work: Über B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen
cites work: Dynamical Gröbner bases
cites work: Making the use of maximal ideals constructive
cites work: Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free
DOI: 10.1016/J.JALGEBRA.2010.04.014
Recommended article: Dynamical Gröbner bases
Recommended article: Q5399018
Recommended article: Q4440634
Recommended article: Q4394968
Recommended article: Q3204585
Recommended article: Gröbner bases in difference-differential modules
Recommended article: Gröbner bases of certain determinantal ideals
Recommended article: Gröbner Bases in D-Modules: Application to Bernstein-Sato Polynomials
Recommended article: Gröbner bases of modules over reduction rings
Recommended article: Some results on Gröbner bases over commutative rings
description: scientific article; zbMATH DE number 5758104