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Gauss: a parameterized domain of computation system with support for signature functions

  • Michael B. Monagan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 722)

Abstract

The fastest known algorithms in classical algebra make use of signature functions. That is, reducing computation with formulae to computing with the integers modulo p, by substituting random numbers for variables, and mapping constants modulo p. This idea is exploited in specific algorithms in computer algebra systems, e.g. algorithms for polynomial greatest common divisors. It is also used as a heuristic to speed up other calculations. But none exploit it in a systematic manner. The goal of this work was twofold. First, to design an AXIOM like system in which these signature functions can be constructed automatically, hence better exploited, and secondly, to exploit them in new ways. In this paper we report on the design of such a system, Gauss.

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References

  1. 1.
    Jenks R., Sutor R.: axiom — The Scientific Computation System, Springer, 1992.Google Scholar
  2. 2.
    Geddes K.O., Labahn G., Czapor S.R.: Algorithms for Computer Algebra Kluwer, 1991.Google Scholar
  3. 3.
    Gonnet G.H.: Determining Equivalence of Expressions in Random Polynomial Time. it Proceedings of the 16th ACM Symposium on the Theory of Computing (1984) 334–341Google Scholar
  4. 4.
    Gonnet G.H.: New Results for Random Determination of Equivalence of Expressions. Proceedings of the 1986 Symposium on Symbolic and Algebraic Computation (1986) 127–131Google Scholar
  5. 5.
    Freeman T., Imirzian G., Kaltofen, E.: DAGWOOD: A System for Manipulating Polynomials Given by Straight-Line Programs. Proceedings of the 1986 Symposium on Symbolic and Algebraic Computation (1986) 169–175Google Scholar
  6. 6.
    Char B.W., Geddes K.O., Gonnet G.H., Leong B.L., Monagan M.B., and Watt S.M.: Maple V Language Reference Manual. Springer-Verlag, New York, 1991.Google Scholar
  7. 7.
    Rabin, M.O.: Probabilistic Algorithm for Testing Primality. J. of Number Theory 12 (1980) 128–138CrossRefGoogle Scholar
  8. 8.
    Solavay, R. Strassen, V.: A fast Monte-Carlo Test for Primality. SIAM J. of Computing 6 (1977) 84–85CrossRefGoogle Scholar
  9. 9.
    Monagan M.B.: Signatures + Abstract Types = Computer Algebra — Intermediate Expression Swell. Ph.D. Thesis, University of Waterloo, 1989.Google Scholar
  10. 10.
    Monagan M.B.: A Heuristic Irreducibility Test for Univariate Polynomials J. Symbolic Comp. 13 No. 1 (1992) 47–57CrossRefGoogle Scholar
  11. 11.
    Schwartz J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM. 27 (1980) 701–717CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Michael B. Monagan
    • 1
  1. 1.Institut für Wissenschaftliches RechnenETH ZentrumZürichSwitzerland

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