Factoring polynomials via relation-finding

  • Victor S. Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 601)


In this paper we describe a new algorithm for fully factoring polynomials defined over the rationals or over number-fields. This algorithm uses as an essential subroutine, any fast relation finding algorithm for vectors of real numbers. Unlike previous algorithms which work on one factor at a time, the new algorithm finds all factors at once. Let P be a polynomial of degree n, height H(P) (=sum of the absolute values of P's coefficients), logarthmic height h(P)=log H(P). If we use the HJLS relation-finding algorithm of Hastad, Just, Lagarias and Schnorr, our algorithm has running time O(n5+Ch(P)) if fast multiplication is used, and O(n6+Ch(P)) if ordinary multiplication is used. This is an improvement by a factor of n over the algorithm of Schönhage, the previously best known.


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  1. [BF89]
    David H. Bailey and Helaman R. P. Ferguson. Numerical results on relations between fundamental constants using a new algorithm. Math. Comp., 53:649–656, 1989.Google Scholar
  2. [FB91]
    Helaman R. P. Ferguson and David H. Bailey. A polynomial time, numerically stable integer relation algorithm. Technical report, Supercomputing Research Center, 17100 Science Drive, Bowie, MD 20715, December 1991.Google Scholar
  3. [GMT89]
    Patrizia Gianni, Victor Miller, and Barry Trager. Decomposition of algebras. In ISAAC '88, Lecture Notes in Computer Science. Springer-Verlag, 1989.Google Scholar
  4. [HJLS89]
    Johann Hastad, B. Just, Jeffrey C. Lagarias, and Claus P. Schnorr. Polynomial time algorithms for finding integer relations among real numbers. Siam J. Comput., 18:859–881, 1989.Google Scholar
  5. [KM86]
    Ravi Kannan and Lyle A. McGeoch. Basis reduction and evidence for transcendence of certain numbers. In Foundations of software technology and theoretical computer science (New Delhi, 1986), volume 241 of Lecture Notes in Comput. Sci., pages 263–269. Springer, Berlin-New York, 1986.Google Scholar
  6. [LLL82]
    Arjen K. Lenstra, Hendrik W. Lenstra, Jr., and Laszlo Lovasz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982.Google Scholar
  7. [Mah64]
    Kurt Mahler. An inequality for the discriminant of a polynomial. Michigan Math. J., 11:257–262, 1964Google Scholar
  8. [Sch82]
    Arnold Schönhage. The fundamental theorem of algebra in terms of computational complexity. Technical report, Math. Inst. Univ. Tübingen, 1982.Google Scholar
  9. [Sch84]
    Arnold Schönhage. Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm. In Automata, languages and programming (Antwerp, 1984), volume 172 of Lecture Notes in Computer Science, pages 436–447, Berlin-New York, 1984. Springer-Verlag.Google Scholar
  10. [Sch87]
    Arnold Schönhage. Equation solving in terms of computational complexity. In Proceedings of the International Congress of Mathematicians, 1986, volume 1, pages 131–153. American Mathematical Society, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Victor S. Miller
    • 1
  1. 1.Thomas J. Watson Research Center Mathematical Sciences DepartmentIBMYorktown Heights

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