Advertisement

Gröbner Bases: A Short Introduction for Systems Theorists

  • Bruno Buchberger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2178)

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.

Keywords

Commutative Algebra Symbolic Computation Short Introduction Univariate Polynomial Multivariate Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W. W. Adams, P. Loustaunau. Introduction to Gröbner Bases. Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 1994.Google Scholar
  2. 2.
    T. Becker, V. Weispfenning. Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, New York, 1993.zbMATHGoogle Scholar
  3. 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. 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].zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    B. Buchberger. Some properties of Grobner bases for polynomial ideals. ACM SIGSAM Bulletin, 10(4):19–24, 1976.CrossRefMathSciNetGoogle Scholar
  6. 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. 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. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 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. 10.
    B. Buchberger. Introduction to Gröbner Bases, pages 3–31 in [9], Cambridge University Press, 1998.Google Scholar
  11. 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. 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. 13.
    S. Collart, M. Kalkbrenner, D. Mall. Converting bases with the Gröbner walk. Journal of Symbolic Computation, 24:465–469, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 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.zbMATHGoogle Scholar
  15. 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.CrossRefMathSciNetGoogle Scholar
  16. 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.CrossRefMathSciNetGoogle Scholar
  17. 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. 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. 19.
    M. Kreuzer and L. Robbiano. Computational Commutative Algebra I. Springer, Heidelberg-New York, 2000. ISBN 3-540-67733-X.zbMATHGoogle Scholar
  20. 20.
    F. Pauer. On lucky ideals for Gröbner basis computations. Journal of Symbolic Computation, 14:471–482, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Franz Winkler. A p-adic approach to the computation of Gröbner bases. Journal of Symbolic Computation, 6:287–304, 1988.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Bruno Buchberger
    • 1
  1. 1.Research Institute for Symbolic ComputationUniversity of LinzSchloss HagenbergAustria

Personalised recommendations