Differential Gröbner bases in one variable and in the partial case
type: Item
instance of: scholarly article
title: Differential Gröbner bases in one variable and in the partial case
zbMATH Open document ID: 0902.13015
DOI: 10.1016/S0895-7177(97)00054-X
author: Giuseppa Carrà Ferro
published in: Mathematical and Computer Modelling
publication date: 1998-12-06T00:00:00Z
review text: Let \(K\) be a field of characteristic zero. Let \(\Delta =\{\delta_1,\ldots,\delta_m\}\) be a set of commuting derivation operators and \(\Theta =\{\delta_1^{a_1}\cdots\delta_m^{a_m}| \;(a_1,\ldots,a_m) \in {\mathbb N}^m\}\) the semigroup of derivatives generated by \(\Delta\). The ring of differential polynomials is defined to be the ring \(K\{Z\} = K[\theta Z| \;\theta\in\Theta]\), where \(Z\) is an indeterminate. This is a commutative polynomial ring with infinite variables. The author [\textit{G. Carrá Ferro}, in: Applied Algebra, algebraic algorithms and error-correcting codes, Proc. 5th Int. Conf., AAECC-5, Menorca 1987, Lect. Notes Comput. Sci. 356, 129-140 (1989; Zbl 0717.13017)] and \textit{F. Ollivier} [ibid., AAECC-8, Lect. Notes Comput. Sci. 508, 304-321 (1991; Zbl 0737.13009)] have introduced the notion of a Gröbner basis for an ideal in \(K\{Z\}\) with respect to a term ordering compatible with the derivations. Differential Gröbner bases do not behave like the ones of ideals in a polynomial ring of finite variables. This paper gives characterizations for differential Gröbner bases. It is also a good survey on this topic with many illustrating examples.
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 1067229
zbMATH Keywords: ring of differential polynomials
zbMATH Keywords: differential Gröbner basis
zbMATH Keywords: differential dimension
reviewed by: Trung Ngo Viet
MaRDI profile type: Publication
cites work: Q5680222
cites work: Associative differential operations
cites work: Q3204585
cites work: Q3975192
cites work: Q4234205
cites work: Q3977389
cites work: Q4038750
full work available at URL: s0895-7177(97)00054-x
OpenAlex ID: W2101707461
description: scientific article; zbMATH DE number 1067229