An Algorithm to Compute Floating Point Groebner Bases

  • Kiyoshi Shirayanagi


Gröbner basis (GB) techniques are a valuable tool for solving many problems in polynomial ideal theory. As is well known, the process of computing a GB may involve large numbers of intermediate coefficients - say from a field k - even when the final GB does not involve many coefficients. In fact, the cost of performing exact arithmetic in k with the intermediate coefficients is a major factor determining the computational cost of computing the GB. This paper proposes a new approach using floating point computation that can be applied when k is a subfield of the real numbers.1


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  1. [1]
    Alefeld, G. and Herzberger, J., Introduction to Interval Computations, Computer Science and Applied Mathematics, Academic Press (1983).Google Scholar
  2. [2]
    Boege, W., Gebauer, R., and Kredel, H., Some Examples for Solving Systems of Algebraic Equations by Calculating Groebner Bases, J. Symb. Comp. 1 (1986), 83–98.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Buchberger, B., Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, Chapter 6 in Multidimensional Systems Theory (N. K. Bose ed.), D. Reidel Publishing Company (1985), 184–232.Google Scholar
  4. [4]
    Buchberger, B., An algorithmical criterion for the solvability of algebraic systems of equations (German), Aequationes mathematicae 4 (3) (1970), 374–383.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., and Watt, S. M., First Leaves: A Tutorial Introduction to Maple V, Springer-Verlag (1992).Google Scholar
  6. [6]
    Dickson, L. E., Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, Am. J. of Math. 35 (1913), 413–426.MATHCrossRefGoogle Scholar
  7. [7]
    Knuth, D. E., The Art of Computer Programming, Vol. 2, Addison-Wesley (1969).Google Scholar
  8. [8]
    Moore, R. E., Interval Arithmetic and Automatic Error Analysis in Digital Computing, Ph.D. Thesis, Mathematics Dept., Stanford Univ., October (1962).Google Scholar
  9. [9]
    Petkovie, M., Iterative Methods for Simultaneous Inclusion of Polynomial Zeros, L. N. Math., Springer-Verlag 1387 (1989).Google Scholar
  10. [10]
    Sasaki, T. and Takeshima, T., A Modular Method for Gröbner-basis Construction over Q and Solving System of Algebraic Equations, J. Information Processing 12 (4) (1989), 371–379.MathSciNetMATHGoogle Scholar
  11. [11]
    Trinks, W., On Improving Approximate Results of Buchberger’s Algorithm by Newton’s Method, SIGSAM Bull. 18 (3) (1984), 7–11.MATHCrossRefGoogle Scholar
  12. [12]
    Winkler, F., A p-adic Approach to the Computation of Gröbner Bases, J. Symb. Comp. 6/2, 3 (1988), 287–304.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Kiyoshi Shirayanagi
    • 1
  1. 1.NTT Communication Science LaboratoriesKyotoJapan

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