article

On the construction of Gröbner bases using syzygies

Author:

H. Michael MöllerAuthors Info & Claims

Journal of Symbolic Computation, Volume 6, Issue 2-3

Pages 345 - 359

https://doi.org/10.1016/S0747-7171(88)80052-X

Published: 01 December 1988 Publication History

Abstract

Grobner bases are useful for analysing multivariate polynomial ideals. For different coefficient domains R, it is shown how to construct (weak) Grobner bases using bases of modules of syzygies, and under constructibility conditions on R an algorithm for finding the required module bases is given. These methods are described in detail for principal ideal rings R. This leads to strong Grobner bases and in case of fields R the construction is the known Buchberger algorithm.

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Cited By

Hofstadler CVerron T(2023)Signature Gröbner bases in free algebras over ringsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597071(298-306)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597071
Ceria MMora F(2022)De Nugis Groebnerialium 6: Rump, Ufnarovski, ZachariasApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-022-00583-233:6(725-749)Online publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1007/s00200-022-00583-2
Francis MVerron TChyzak FLabahn G(2021)On Two Signature Variants of Buchberger's Algorithm over Principal Ideal DomainsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465522(139-146)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465522
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Index Terms

On the construction of Gröbner bases using syzygies
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness

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cover image Journal of Symbolic Computation

Journal of Symbolic Computation Volume 6, Issue 2-3

Oct./Dec. 1988

247 pages

ISSN:0747-7171

Editor:
Bruno Buchberger

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Copyright © Academic Press Limited © 1988.

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Academic Press, Inc.

United States

Publication History

Published: 01 December 1988

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Cited By

Hofstadler CVerron T(2023)Signature Gröbner bases in free algebras over ringsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597071(298-306)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597071
Ceria MMora F(2022)De Nugis Groebnerialium 6: Rump, Ufnarovski, ZachariasApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-022-00583-233:6(725-749)Online publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1007/s00200-022-00583-2
Francis MVerron TChyzak FLabahn G(2021)On Two Signature Variants of Buchberger's Algorithm over Principal Ideal DomainsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465522(139-146)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465522
Ceria MMora T(2020)Toward involutive bases over effective ringsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-020-00448-631:5-6(359-387)Online publication date: 1-Nov-2020
https://dl.acm.org/doi/10.1007/s00200-020-00448-6
Barkee BCeria MMoriarty TVisconti A(2020)Why you cannot even hope to use Gröbner bases in cryptography: an eternal golden braid of failuresApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-020-00428-w31:3-4(235-252)Online publication date: 1-Jun-2020
https://dl.acm.org/doi/10.1007/s00200-020-00428-w
Hashemi AKazemi M(2019)Parametric Standard Bases and Their ApplicationsComputer Algebra in Scientific Computing10.1007/978-3-030-26831-2_13(179-196)Online publication date: 26-Aug-2019
https://dl.acm.org/doi/10.1007/978-3-030-26831-2_13
Dehghani Darmian MHashemi A(2017)Parametric FGLM algorithmJournal of Symbolic Computation10.1016/j.jsc.2016.12.00682:C(38-56)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1016/j.jsc.2016.12.006
Markwig TRen YWienand O(2017)Standard bases in mixed power series and polynomial rings over ringsJournal of Symbolic Computation10.1016/j.jsc.2016.08.00979:P1(119-139)Online publication date: 1-Mar-2017
https://dl.acm.org/doi/10.1016/j.jsc.2016.08.009
Mora FRobertz DLinton SYokoyama K(2015)De Nugis Groebnerialium 4Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756640(283-290)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756640
Gopalakrishnan SKalla PMeredith MEnescu FGielen G(2007)Finding linear building-blocks for RTL synthesis of polynomial datapaths with fixed-size bit-vectorsProceedings of the 2007 IEEE/ACM international conference on Computer-aided design10.5555/1326073.1326103(143-148)Online publication date: 5-Nov-2007
https://dl.acm.org/doi/10.5555/1326073.1326103
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