Some results on Gröbner bases over commutative rings
type: Item
instance of: scholarly article
title: Some results on Gröbner bases over commutative rings
zbMATH Open document ID: 0767.13008
DOI: 10.1016/S0747-7171(10)80007-0
author: William W. Adams
author: Ann K. Boyle
published in: Journal of Symbolic Computation
publication date: 1993-01-16T00:00:00Z
review text: Let \(R\) be a commutative ring with identity. Fix a term order on the set of monomials in \(R[x_ 1,\ldots,x_ m]\) (the ring of polynomials). A set \(F \subseteq R[x_ 1,\ldots,x_ m]\) is defined as usual a Gröbner basis (with coefficients in \(R\)) if \(\text{Lt} (F)=\text{Lt}(\langle F \rangle)\), i.e. if the ideal generated by the leading terms of the polynomials in \(F\) coincides with the set (ideal) of all the terms of the polynomials of the ideal generated by \(F\). Fix now two sets of variables: \(y:=\{y_ 1,\ldots,y_ n\}\) and \(x:=\{x_ 1,\ldots,x_ m\}\) and let \(R[x,y]\) denote the polynomial ring in the variables \(x\) and \(y\) respectively. Assume that the monomials in \(y\) (in \(x\)) have a term order denoted by \(<_ y\) (\(<_ x\) resp.); then one can define a term order on the monomials in \(y\) and \(x\) by: \[ y_ 1x_ 1 < y_ 2x_ 2 \quad\text{if}\quad x_ 1<_ x x_ 2 \;\text{ or } \;\;x_ 1 = x_ 2 \;\;\text{ and } \;y_ 1<_ y y_ 2, \] where \(y_ 1,y_ 2\) (\(x_ 1,x_ 2\)) are monomials in \(y\) (in \(x\) resp.). Let \(F:=\{f_ 1,\ldots,f_ t\}\subseteq R[x,y]\). Set \(G:=\{\text{lc}_ x(f_ 1), \ldots,\text{lc}_ x(f_ t) \}\), where \(\text{lc}_ x(f)\) is the leading coefficient of \(f\) as a polynomial with coefficients in \(R[y]\) with respect to the term order \(<_ x\). The paper analyzes the following problems: When is \(F\) a Gröbner basis with coefficients in \(R\)? When is \(F\) a Gröbner basis with coefficients in \(R[y]\)? When is \(G\) a Gröbner basis in \(R[y]\) with coefficients in \(R\)? Several results concerning these problems are established. In particular: Assume \(m=1\), i.e. \(x=\{x_ 1\}\). Then \(F\) is a Gröbner basis in \(R[x,y]\) with coefficients in \(R\) if and only if the following two conditions are satisfied: (i) \(F\) is a Gröbner basis in \(R[x,y]\) with coefficients in \(R[x]\) and (ii) for every \(i=0,\ldots,s\), \({\mathcal G}_ i\) is a Gröbner basis in \(R[y]\) with coefficients in \(R\) (where \({\mathcal G}_ i:= \{\text{lc}_ x…
reviewed by: Alessandro Logar
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 92592
zbMATH Keywords: Gröbner basis
zbMATH Keywords: term order on the monomials
zbMATH Keywords: Szekeres basis
zbMATH Keywords: UFD
MaRDI profile type: Publication
cites work: On constructing bases for ideals in polynomial rings over the integers
cites work: A theorem on refining division orders by the reverse lexicographic order
cites work: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal
cites work: Ideal basis and primary decompositions: case of two variables
cites work: On the construction of Gröbner bases using syzygies
cites work: Lifting canonical algorithms from a ring R to the ring R[x]
cites work: A Canonical Basis for the Ideals of a Polynomial Domain
full work available at URL: s0747-7171(10)80007-0
OpenAlex ID: W2032144352
description: scientific article; zbMATH DE number 92592