SAGBI and SAGBI-Gröbner bases over principal ideal domains
type: Item
instance of: scholarly article
title: SAGBI and SAGBI-Gröbner bases over principal ideal domains
zbMATH Open document ID: 0930.13021
author: William W. Adams
author: Serkan Hoşten
author: Philippe Loustaunau
author: J. Lyn Miller
published in: Journal of Symbolic Computation
publication date: 1999-12-01T00:00:00Z
review text: SAGBI bases (``Subalgebra Analogues of Gröbner Bases for Ideals'') were introduced by \textit{L. Robbiano} and \textit{M. Sweedler} [in: Commutative Algebra, Proc. Workshop, Salvador 1988, Lect. Notes Math. 1430, 61-87 (1988; Zbl 0725.13013)]. They are defined as follows: Let \(A:=R[f_1,\dots,f_m]\subseteq R[x_1,\dots,x_n]\) be the algebra generated over \(R\) by the polynomials \(f_1,\dots f_m\) and assume a term-order is fixed in \(R[x_1,\dots,x_n]\). Then a subset \(S\) of \(A\) is a SAGBI basis for \(A\) if \(R[Lt(S)]=R[Lt(A)]\) (where \(Lt(S)\) and \(Lt(A)\) denote the set of leading terms of the elements of \(S\) and of \(A\) respectively). The above definition is usually given in the case where \(R\) is a field, but in the present paper, the authors are mainly interested in the more general case in which \(R\) is a principal ideal domain. First of all they discuss a Buchberger-type algorithm for the computation of SAGBI bases (when SAGBI bases exist, which is not always the case). Successively they study SAGBI-Gröbner bases for ideals of \(A\) (which are bases with properties similar to Gröbner bases of ideals in polynomials rings). Assuming that \(A\) has a finite SAGBI bases, they give an algorithm for computing SAGBI-Gröbner bases. In the last section they show that strong SAGBI bases -- the analogue of strong Gröbner bases for ideals in polynomial rings -- always exist (under the assumption that a finite SAGBI basis exists) and can be computed with an algorithm. This construction furthermore implies some results in integer programming.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1271791
zbMATH Keywords: Gröbner bases
zbMATH Keywords: SAGBI bases
zbMATH Keywords: SAGBI-Gröbner bases
zbMATH Keywords: integer programming
zbMATH Keywords: Buchberger algorithm
reviewed by: Alessandro Logar
MaRDI profile type: Publication
OpenAlex ID: W2075381343
DOI: 10.1006/JSCO.1998.0243
description: scientific article; zbMATH DE number 1271791