Gröbner bases and normal forms in a subring of the power series ring on countably many variables
type: Item
instance of: scholarly article
title: Gröbner bases and normal forms in a subring of the power series ring on countably many variables
zbMATH Open document ID: 0929.13015
author: J. E. Snellman
published in: Journal of Symbolic Computation
publication date: 2000-01-24T00:00:00Z
review text: Let \(K\) be a field and let \(R=Kx_1,x_2,x_3,\dots\) be the ring of power series in countably many variables, with coefficients in \(K\). Denote by \(R'\) the smallest \(K\)-subalgebra of \(R\) that contains all homogeneous elements. The ring \(R'\) is of interest also because it allows a generalization of the notion of generic form in the polynomial ring \(K[x_1,\dots,x_n]\). Ideals generated by generic forms have been the focus of much study, and the investigation of their initial ideals leds to the study of Gröbner bases theory of the ring \(R'\), that is the topic of this paper. The author generalizes the notion of admissible term order to this context and defines the leading power product of an element of \(R'\), and thus the initial ideal \(\text{gr}(I)\) of an arbitrary ideal \(I\subset R'\). It turns out that if \(I\) is locally finitely generated, then \(\text{gr}(I)\) is also locally finitely generated; this implies that \(I\) has a finite truncated Gröbner basis up to any total degree. An example is given of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic order.
reviewed by: Emanuela De Negri
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1204278
zbMATH Keywords: Gröbner bases
zbMATH Keywords: generic forms
zbMATH Keywords: power series ring
zbMATH Keywords: homogeneous elements
zbMATH Keywords: admissible term order
MaRDI profile type: Publication
OpenAlex ID: W2125497504
DOI: 10.1006/JSCO.1997.0178
description: scientific article; zbMATH DE number 1204278