Probabilistic algorithms for sparse polynomials

  • Richard Zippel
6. Polynomial Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 72)


In this paper we have tried to demonstrate how sparse techniques can be used to increase the effectiveness of the modular algorithms of Brown and Collins. These techniques can be used for an extremely wide class of problems and can applied to a number of different algorithms including Hensel's lemma. We believe this work has finally laid to rest the bad zero problem.

Much of the work here is the direct result of discussion with Barry Trager and Joel Moses whose help we wish to acknowledge.


Nonzero Coefficient Exponential Behavior Probabilistic Algorithm Multivariate Polynomial Zero Coefficient 
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.


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  1. 1.
    S. Brown, “On Euclid's Algorithm and the Computation of Polynomial Greatest Divisors,” J. ACM 18, 4 (1971), 478–504.Google Scholar
  2. 2.
    MATHLAB Group, MACSYMA Reference Manual—version 9, Laboratory for Computer Science, Massachusetts Institute of Technology, (1978).Google Scholar
  3. 3.
    J. Moses and D. Y. Y. Yun, “The EZGCD algorithm,” Proceedings of ACM Nat. Conf. (1973), 159–166.Google Scholar
  4. 4.
    D. R. Musser, “Multivariate Polynomial Factoring,” J. ACM 22, 2 (1975), 291–308.Google Scholar
  5. 5.
    M. O. Rabin, “Probabilistic Algorithms,” Algorithms and Complexity—New Directions and Recent Results (J. F. Traub Ed.), Acad. Press, New York, (1976), 21–39.Google Scholar
  6. 6.
    R. Solovay and V. Strassen, “A Fast Monte-Carlo Test for Primality,” SIAM J. of Comp. 6, 1 (1977).Google Scholar
  7. 7.
    P. S.-H. Wang and L. P. Rothschild, “Factoring Multivariate Polynomials over the Integers,” Math. Comp. 29, (1975), 935–950.Google Scholar
  8. 8.
    P. S.-H. Wang, “An Improved Multivariate Polynomial Factoring Algorithm,” Math. Comp. 32, (1978), 1215–1231.Google Scholar
  9. 9.
    D. Y. Y. Yun, The Hensel Lemma in Algebraic Manipulation, Ph. D. thesis, Massachusetts Institute of Technology, (1974).Google Scholar
  10. 10.
    H. Zassenhaus, “On Hensel Factorization I,” J. Number Theory 1, (1969), 291–311.Google Scholar
  11. 11.
    R. E. Zippel, Probabilistic Algorithms for Sparse Polynomials, Ph. D. thesis, Massachusetts Institute of Technology, (1979).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • Richard Zippel
    • 1
  1. 1.Laboratory for Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA

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