Advertisement

Usage of Modular Techniques for Efficient Computation of Ideal Operations

(Invited Talk)
  • Kazuhiro Yokoyama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

Modular techniques are widely applied to various algebraic computations. (See [5] for basic modular techniques applied to polynomial computations.) In this talk, we discuss how modular techniques are efficiently applied to computation of various ideal operations such as Gröbner base computation and ideal decompositions. Here, by modular techniques we mean techniques using certain projections for improving the efficiency of the total computation, and by modular computations, we mean corresponding computations applied to projected images.

References

  1. 1.
    Arnold, E.: Modular algorithms for computing Gröbner bases. J. Symb. Comp. 35, 403–419 (2003)zbMATHCrossRefGoogle Scholar
  2. 2.
    Dahan, X., Kadri, A., Schost, É.: Bit-size estimates for triangular sets in positive dimension. J. Complexity 28, 109–135 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dahan, X., Moreno Maza, M., Schost, É., Wu, W., Xie, Y.: Lifting techniques for triangular decompositions. In: Proc. ISSAC 2005, pp. 108–115. ACM Press, New York (2005)Google Scholar
  4. 4.
    Dahan, X., Schost, É.: Sharp estimates for triangular sets. In: Proc. ISSAC 2004, pp. 103–110. ACM Press, New York (2004)Google Scholar
  5. 5.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  6. 6.
    Gräbe, H.: On lucky primes. J. Symb. Comp. 15, 199–209 (1993)zbMATHCrossRefGoogle Scholar
  7. 7.
    Idrees, N., Pfister, G., Steidel, S.: Parallelization of modular algorithms. J. Symb. Comp. 46, 672–684 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Orange, S., Renault, G., Yokoyama, K.: Efficient arithmetic in successive algebraic extension fields using symmetries. Math. Comput. Sci. (to appear)Google Scholar
  9. 9.
    Noro, M., Yokoyama, K.: A modular method to compute the rational univariate representation of zero-dimensional ideals. J. Symb. Comp. 28, 243–263 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Noro, M., Yokoyama, K.: Implementation of prime decomposition of polynomial ideals over small finite fields. J. Symb. Comp. 38, 1227–1246 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pauer, F.: On lucky ideals for Gröbner bases computations. J. Symb. Comp. 14, 471–482 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Renault, G., Yokoyama, K.: Multi-modular algorithm for computing the splitting field of a polynomial. In: Proceedings of ISSAC 2008, pp. 247–254. ACM Press, New York (2008)Google Scholar
  13. 13.
    Sasaki, T., Takeshima, T.: A modular method for Gröbner-bases construction over ℚ and solving system of algebraic equations. J. Inform. Process. 12, 371–379 (1989)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Traverso, C.: Gröbner Trace Algorithms. In: Gianni, P. (ed.) ISSAC 1988. LNCS, vol. 358, pp. 125–138. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  15. 15.
    Winkler, F.: A p-adic approach to the computation of Gröbner bases. J. Symb. Comp. 6, 287–304 (1988)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kazuhiro Yokoyama
    • 1
  1. 1.Department of MathematicsRikkyo UniversityToshima-kuJapan

Personalised recommendations