Advertisement

The computation of polynomial greatest common divisors over an algebraic number field

  • Lars Langemyr
  • Scott McCallum
Polynomial Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 378)

Abstract

We present an algorithm for computing the greatest common divisor of two polynomials over an algebraic number field. We obtain a better computing time bound for this algorithm than for previously published algorithms solving the same problem. We have also performed empirical run time tests which have confirmed this. Our motivation for seeking the algorithm of the present paper stems from our interest in the cylindrical algebraic decomposition (cad) algorithm (Collins [3], Arnon et al. [1]). The cad algorithm makes essential use of algebraic polynomial gcd computations, which often dominate the cost of the algorithm.

Our algorithm is a generalization of the modular algorithm developed by Brown

References

  1. [1]
    D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition. SIAM Journal on Computing, 13:865–877, 878–889, 1984.CrossRefGoogle Scholar
  2. [2]
    W. S. Brown. On Euclid's algorithm and the computation of polynomial greatest common divisors. Journal of the ACM, 18(4):478–504, October 1971.CrossRefGoogle Scholar
  3. [3]
    G. E. Collins. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, pages 134–183. Volume 33 of Lecture Notes in Computer Science, Springer-Verlag, 1975.Google Scholar
  4. [4]
    G. E. Collins. The SAC-1 Polynomial GCD and Resultant System. Technical Report University of Wisconsin, MACC Tech. Report 27, University of Wisconsin, Madison, 1972.Google Scholar
  5. [5]
    S. Landau. Factoring polynomials over algebraic number fields. SIAM Journal on Computing, 14:184–195, 1985.CrossRefGoogle Scholar
  6. [6]
    L. Langemyr. Computing the GCDs of Polynomials Over Algebraic Number Fields. PhD thesis, Royal Institute of Technology, 1988. to appear.Google Scholar
  7. [7]
    C. M. Rubald. Algorithms for polynomials over a real algebraic number field. PhD thesis, University of Wisconsin, Madison, January 1974.Google Scholar
  8. [8]
    P. J. Weinberger and L. P. Rothschild. Factoring polynomials over algebraic number fields. ACM Transactions on Mathematical Software, 2(4):335–350, December 1976.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Lars Langemyr
    • 1
  • Scott McCallum
    • 2
  1. 1.NADA, Royal Institute of TechnologyStockholmSweden
  2. 2.Research School of Physical SciencesThe Australian National UniversityCanberraAustralia

Personalised recommendations