Skip to main content
SpringerLink
Log in

Gröbner Bases: A Short Introduction for Systems Theorists

  • Conference paper
  • First Online:
Book cover Computer Aided Systems Theory — EUROCAST 2001 (EUROCAST 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2178))

Included in the following conference series:

Abstract

In this paper, we give a brief overview on Gröbner bases theory, addressed to novices without prior knowledge in the field. After explaining the general strategy for solving problems via the Gröbner approach, we develop the concept of Gröbner bases by studying uniquenss of polynomial division (“reduction”). For explicitly constructing Gröbner bases, the crucial notion of S—polynomials is introduced, leading to the complete algorithmic solution of the construction problem. The algorithm is applied to examples from polynomial equation solving and algebraic relations. After a short discussion of complexity issues, we conclude the paper with some historical remarks and references.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. W. Adams, P. Loustaunau. Introduction to Gröbner Bases. Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 1994.

    Google Scholar 

  2. T. Becker, V. Weispfenning. Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, New York, 1993.

    MATH  Google Scholar 

  3. B. Buchberger. An Algorithm for Finding the Bases Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal (German). PhD thesis, Univ. of Innsbruck (Austria), 1965.

    Google Scholar 

  4. B. Buchberger. An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations (German). Aequationes Mathematicae, 4(3):374–383, 1970. English translation in [9].

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Buchberger. Some properties of Grobner bases for polynomial ideals. ACM SIGSAM Bulletin, 10(4):19–24, 1976.

    Article  MathSciNet  Google Scholar 

  6. B. Buchberger. A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner Bases. In Edward W. Ng, editor, Proceedings of the International Symposium on Symbolic and Algebraic Manipulation (EUROSAM’ 79), Marseille, France, volume 72 of Lecture Notes in Computer Science, pages 3–21. Springer, 1979.

    Google Scholar 

  7. B. Buchberger. Gröbner-Bases: An Algorithmic Method in Polynomial Ideal Theory. In N. K. Bose, editor, Multidimensional Systems Theory, chapter 6, pages 184–232. Reidel Publishing Company, Dodrecht, 1985.

    Google Scholar 

  8. B. Buchberger and J. Elias. Using Gröbner Bases for Detecting Polynomial Identities: A Case Study on Fermat’s Ideal. Journal of Number Theory 41:272–279, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  9. B. Buchberger and F. Winkler, editors. Gröbner Bases and Applications, volume 251 of London Mathematical Society Series. Cambridge University Press, 1998. Proc. of the International Conference “33 Years of Groebner Bases”.

    Google Scholar 

  10. B. Buchberger. Introduction to Gröbner Bases, pages 3–31 in [9], Cambridge University Press, 1998.

    Google Scholar 

  11. B. Buchberger. Gröbner-Bases and System Theory. To appear as Special Issue on Applications of Gröbner Bases in Multidimensional Systems and Signal Processing, Kluwer Academic Publishers, 2001.

    Google Scholar 

  12. A. Capani, G. Niesi, and L. Robbiano. CoCoA: A System for Doing Computations in Commuatative Algebra, 1998. Available via anonymous ftp from http://cocoa.dima.uniqe.it.

  13. S. Collart, M. Kalkbrenner, D. Mall. Converting bases with the Gröbner walk. Journal of Symbolic Computation, 24:465–469, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Cox, J. Little, D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, New York, 1992.

    MATH  Google Scholar 

  15. L. E. Dickson. Finiteness of the Odd Perfect and Primitive Abundant Numbers within Distince Prime Factors. American Journal of Mathematics, 35:413–426, 1913.

    Article  MathSciNet  Google Scholar 

  16. J. C. Faugére, P. Gianni, D. Lazard, T. Mora. Efficient computation of zero-dimensional Gröbner bases by change of ordering. Journal of Symbolic Computation, 16:377–399, 1993.

    Article  MathSciNet  Google Scholar 

  17. D. Grayson and M. Stillman. Macaulay 2: A Software System for Algebraic Geometry and Commutative Algebra. Available over the web at http://www.math.uiuc.edu/-Macaulay2.

  18. G.-M. Greuel and G. Pfister and H. Schönemann. Singular Reference Manual. Reports On Computer Algebra, Number 12, Centre for Computer Algebra, University of Kaiserslautern, 1997. Available over the web at http://www.mathematik.uni?kl.de/~zca/Singular.

  19. M. Kreuzer and L. Robbiano. Computational Commutative Algebra I. Springer, Heidelberg-New York, 2000. ISBN 3-540-67733-X.

    MATH  Google Scholar 

  20. F. Pauer. On lucky ideals for Gröbner basis computations. Journal of Symbolic Computation, 14:471–482, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  21. Franz Winkler. A p-adic approach to the computation of Gröbner bases. Journal of Symbolic Computation, 6:287–304, 1988.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Symbolic Computation, University of Linz, A4232, Schloss Hagenberg, Austria

    Bruno Buchberger

Authors
  1. Bruno Buchberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Universidad de Las Palmas de Gran Canaria, Instituto Universitario de Ciencias y Tecnologías Cibernéticas, Campus de Tafira, 35017, Las Palmas de Gran Canaria, Spain

    Roberto Moreno-Díaz

  2. University of Linz, Research Institute for Symbolic Computation, 232, Schloss Hagenberg, Austria

    Bruno Buchberger

  3. University ofA Coruña, Facultad de Informática, LFCIA, Dpto. of Computer Science, Campus de Elviña, 15071, A Coruña, Spain

    José Luis Freire

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Buchberger, B. (2001). Gröbner Bases: A Short Introduction for Systems Theorists. In: Moreno-Díaz, R., Buchberger, B., Luis Freire, J. (eds) Computer Aided Systems Theory — EUROCAST 2001. EUROCAST 2001. Lecture Notes in Computer Science, vol 2178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45654-6_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-45654-6_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42959-3

  • Online ISBN: 978-3-540-45654-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation