Reduced Gröbner bases, free difference-differential modules and difference-differential dimension polynomials
type: Item
instance of: scholarly article
title: Reduced Gröbner bases, free difference-differential modules and difference-differential dimension polynomials
zbMATH Open document ID: 0994.13009
author: Alexander B. Levin
published in: Journal of Symbolic Computation
publication date: 2001-03-11T00:00:00Z
review text: Difference dimension polynomials play the same role in difference algebra as Hilbert polynomials in commutative algebra or differential dimension polynomials in differential algebra. The paper extends the notion of difference dimension polynomials and differential dimension polynomials introducing the notion of difference-differential polynomials associated with finitely generated difference-differential field extensions. In order to study these dimension polynomials the author defines a notion of reduction in free modules over rings of difference-differential operators which allows to develop an analog of Gröbner basis theory. One of the main theorems of the paper generalizes both Kolchin's theorem on dimension polynomials of differential field extensions and the corresponding theorem for difference fields due to the author. The developed theory is then applied to get a method for computing the dimension polynomial associated with a system of linear difference-differential equations.
reviewed by: Alessandro Logar
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 12H10
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 1538749
zbMATH Keywords: difference-differential modules
zbMATH Keywords: difference-differential dimension polynomials
zbMATH Keywords: difference-differential polynomials
zbMATH Keywords: difference-differential field extensions
zbMATH Keywords: Gröbner basis
MaRDI profile type: Publication
full work available at URL: jsco.1999.0412
OpenAlex ID: W2024284700
cites work: A Gröbner approach to involutive bases
cites work: The theory of involutive divisions and an application to Hilbert function computations
cites work: Dimension polynomials of extensions of difference fields
cites work: Q4693774
cites work: Q4313087
cites work: Q3204585
cites work: Differential Gröbner bases in one variable and in the partial case
cites work: Q5334359
cites work: Q4317713
cites work: Finitely Generated Extensions of Partial Difference Fields
cites work: Q4394968
cites work: Differential Dimension Polynomials and a Fundamental Theorem on Differential Modules
cites work: Kahler Differentials and Differential Algebra in Arbitrary Characteristic
cites work: On the Differential Transcendence Polynomials of Finitely Generated Differential Field Extensions
cites work: The notion of dimension in the theory of algebraic differential equations
cites work: Q5680222
cites work: COMPUTATION OF DIMENSION POLYNOMIALS
cites work: Q4229400
cites work: CHARACTERISTIC POLYNOMIALS OF FILTERED DIFFERENCE MODULES AND OF EXTENSIONS OF DIFFERENCE FIELDS
cites work: CHARACTERISTIC POLYNOMIALS OF INVERSE DIFFERENTIAL MODULES AND SOME PROPERTIES OF INVERSE DIFFERENTIAL DIMENSION
cites work: Q3209214
cites work: A generalization of Gröbner basis algorithms to polycyclic group rings
cites work: Q3963117
cites work: Q4002365
cites work: Q3695369
cites work: Q4394975
cites work: Computations in differential and difference modules
cites work: Galois theory of difference equations
cites work: Q5589310
cites work: Well-Ordering of Certain Numerical Polynomials
cites work: Differential Dimension Polynomials of Finitely Generated Extensions
cites work: Q4344108
cites work: Q5184913
DOI: 10.1006/JSCO.1999.0412
Recommended article: Gröbner bases in difference-differential modules and difference-differential dimension polynomials
Recommended article: Gröbner bases in difference-differential modules
Recommended article: Computing difference-differential dimension polynomials by relative Gröbner bases in difference-differential modules
Recommended article: Gröbner bases with respect to several orderings on difference-differential modules and multivariate dimension polynomials
Recommended article: Gröbner bases in difference-differential modules with coefficients in a commutative ring
Recommended article: Q4314653
Recommended article: Q5199089
Recommended article: Computation of dimension in filtered free modules by Gröbner reduction
Recommended article: Q3204585
Recommended article: Q3175736
description: scientific article; zbMATH DE number 1538749