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Reduced Gröbner Bases, Free Difference-Differential Modules and Difference-Differential Dimension Polynomials

Author:

Alexander LevinAuthors Info & Claims

Journal of Symbolic Computation, Volume 30, Issue 4

Pages 357 - 382

https://doi.org/10.1006/jsco.1999.0412

Published: 01 October 2000 Publication History

Abstract

We define a special type of reduction in a free left module over a ring of difference differential operators and use the idea of the Gr bner basis method to develop a technique that allows us to determine the Hilbert function of a finitely generated difference differential module equipped with the natural double filtration. The results obtained are applied to the study of difference differential field extensions and systems of difference differential equations. We prove a theorem on difference differential dimension polynomial that generalizes both the classical Kolchin s theorem on dimension polynomial of a differential field extension and the corresponding author s result for difference fields. We also determine invariants of a difference differential dimension polynomial and consider a method of computation of the dimension polynomial associated with a system of linear difference differential equations.

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Cited By

Levin AChyzak FLabahn G(2021)Generalized Gröbner Bases and New Properties of Multivariate Difference Dimension PolynomialsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465544(273-280)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465544
Levin AKauers MOvchinnikov ASchost É(2018)Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals.Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209008(255-262)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209008
Fürst CLandsmann GRobertz DLinton SYokoyama K(2015)Computation of Dimension in Filtered Free Modules by Gröbner ReductionProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756680(181-188)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756680
Show More Cited By

Index Terms

Reduced Gröbner Bases, Free Difference-Differential Modules and Difference-Differential Dimension Polynomials
1. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical analysis

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Information & Contributors

Information

Published In

cover image Journal of Symbolic Computation

Journal of Symbolic Computation Volume 30, Issue 4

Special issue on applications of the Gröbner basis method

Oct. 2000

151 pages

ISSN:0747-7171

Editors:
Hoon Hong
North Carolina State Univ., Raleigh
,
Quoc-Nam Tran
Lamar Univ., Beaumont, TX
,
Franz Winkler
Johannes Kepler Univ., Linz, Austria

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Copyright © Academic Press.

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Academic Press, Inc.

United States

Publication History

Published: 01 October 2000

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Cited By

Levin AChyzak FLabahn G(2021)Generalized Gröbner Bases and New Properties of Multivariate Difference Dimension PolynomialsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465544(273-280)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465544
Levin AKauers MOvchinnikov ASchost É(2018)Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals.Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209008(255-262)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209008
Fürst CLandsmann GRobertz DLinton SYokoyama K(2015)Computation of Dimension in Filtered Free Modules by Gröbner ReductionProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756680(181-188)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756680
Levin A(2015)Dimension Polynomials of Intermediate Fields of Inversive Difference Field ExtensionsRevised Selected Papers of the 6th International Conference on Mathematical Aspects of Computer and Information Sciences - Volume 958210.1007/978-3-319-32859-1_31(362-376)Online publication date: 11-Nov-2015
https://dl.acm.org/doi/10.1007/978-3-319-32859-1_31
Levin AMonagan MCooperman GGiesbrecht M(2013)Multivariate difference-differential dimension polynomials and new invariants of difference-differential field extensionsProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465521(267-274)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1145/2465506.2465521
Zhou MWinkler F(2008)Computing difference-differential dimension polynomials by relative Gröbner bases in difference-differential modulesJournal of Symbolic Computation10.1016/j.jsc.2008.02.00143:10(726-745)Online publication date: 1-Oct-2008
https://dl.acm.org/doi/10.1016/j.jsc.2008.02.001
Levin AWang D(2007)Gröbner bases with respect to several term orderings and multivariate dimension polynomialsProceedings of the 2007 international symposium on Symbolic and algebraic computation10.1145/1277548.1277583(251-260)Online publication date: 29-Jul-2007
https://dl.acm.org/doi/10.1145/1277548.1277583
Zhou MWinkler FTrager B(2006)Gröbner bases in difference-differential modulesProceedings of the 2006 international symposium on Symbolic and algebraic computation10.1145/1145768.1145825(353-360)Online publication date: 9-Jul-2006
https://dl.acm.org/doi/10.1145/1145768.1145825

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