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A unified method for multivariate polynomial factorizations

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Abstract

Recently, Sasaki et al. presented an approximate factorization algorithm of multivariate polynomials. The algorithm calculates irreducible factors by investigating linear combinations of the same power of approximate roots. In this paper, we show that various kinds of multivariate polynomial factorizations can be performed by this method. We present algorithms for factorization of multivariate polynomials over power-series rings, over the integers, over algebraic number fields including algebraically closed fields, and over algebraic function fields. Furthermore, we discuss applicability of this method to univariate polynomial factorization.

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References

  • [Ber67] E.R. Berlekamp, Factorizing polynomials over finite fields. Bell System Tech. J.,46 (1967), 1853–1859.

    MathSciNet  Google Scholar 

  • [ChGr82] A.L. Christov and D.Y. Grigor’ev, Polynomial-time factoring of multivariate polynomials over global field. LOMI preprint, Leningrad, 1882.

  • [HeSi81] J. Heintz and M. Sieveking, Absolute primality of polynomials is decidable in random polynomial time in the number of variables. Lecture Notes in Comp. Sci., Vol. 115 (Proc. 1981 Intern. Conf. on Automata, Languages and Programming), Springer-Verlag, Berlin-Heidelberg-New York, 1981, 16–28.

    Google Scholar 

  • [Kal82] E. Kaltofen, A polynomial reduction from multivariate to bivariate integral polynomial factorization. Proc. 1982 ACM Symp. on Theory of Computation, 1982, ACM, New York, 261–266.

  • [Kal85] E. Kaltofen, Fast parallel absolute irreducibility testing. J. Symb. Comput.,1 (1985), 57–67.

    Article  MATH  MathSciNet  Google Scholar 

  • [Len82] A.K. Lenstra, Lattice and factorization of polynomials over algebraic number fields. Lecture Notes in Comp. Sci., Vol. 144 (Proc. EUROCAM ’82), Springer-Verlag, Berlin-Heidelberg-New York, 1982, 32–39.

    Google Scholar 

  • [Len84] A.K. Lenstra, Factoring multivariate polynomials over algebraic number field. Lecture Notes in Comp. Sci., Vol. 176, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984, 389–396.

    Google Scholar 

  • [LLL82] A.K. Lenstra, Jr., H.W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann.,261 (1982), 515–534.

    Article  MATH  MathSciNet  Google Scholar 

  • [SSH92] T. Sasaki, T. Saito and T. Hilano, Analysis of approximate factorization algorithm, I. Japan J. Indust. Appl. Math.,9 (1992), 351–368.

    Article  MATH  MathSciNet  Google Scholar 

  • [SSKS91] T. Sasaki, M. Suzuki, M. Kolář and M. Sasaki, Approximate factorization of multivariate polynomials and absolute irreducibility testing. Japan J. Indust. Appl. Math.,8 (1991), 357–375.

    MATH  MathSciNet  Google Scholar 

  • [Tra76] B.M. Trager, Algebraic factoring and rational function integration. Proc. ACM Symp. on Symbolic and Algebraic Computation, 1976, ACM, New York, 219–226.

  • [vzGK83] J. von zur Gathen and E. Kaltofen, A polynomial-time algorithm for factoring multivariate polynomials over finite fields. Lecture Notes in Comp. Sci., Vol. 154 (Proc. 1983 Intern. Conf. on Automata, Languages and Programming), Springer-Verlag, Berlin-Heidelberg-New York, 1983, 250–263.

    Google Scholar 

  • [Wan76] P.S. Wang, Factoring multivariate polynomials over algebraic number fields. Math. Comp.,30 (1976), 324–336.

    Article  MATH  MathSciNet  Google Scholar 

  • [WaRo75] P.S. Wang and L.P. Rothschild, Factoring multivariate polynomials over the integers. Math. Comp.,29 (1975), 935–950.

    Article  MATH  MathSciNet  Google Scholar 

  • [WeRo76] P.J. Weinberger and L.P. Rothschild, Factoring polynomials over algebraic number fields. ACM Trans. Math. Software,2 (1976), 335–350.

    Article  MATH  MathSciNet  Google Scholar 

  • [YNT90] K. Yokoyama, M. Noro and T. Takeshima, On factoring multi-variate polynomials over algebraically closed fields. Proc. 1990 Intern. Symp. on Symbolic and Algebraic Computation, ACM, New York, 1990, 297.

  • [Zas69] H. Zassenhaus, On Hensel factorization I. J. Number Theory,1 (1969), 291–311.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, 305, Tsukuba-shi, Ibaraki, Japan

    Tateaki Sasaki

  2. The Institute of Physical and Chemical Research (RIKEN), 351-01, Wako-shi, Saitama, Japan

    Mutsuko Sasaki

Authors
  1. Tateaki Sasaki
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  2. Mutsuko Sasaki
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Additional information

Work supported in part by Japanese Ministry of Education, Science and Culture under Grants 03558008 and 04245102.

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Sasaki, T., Sasaki, M. A unified method for multivariate polynomial factorizations. Japan J. Indust. Appl. Math. 10, 21–39 (1993). https://doi.org/10.1007/BF03167201

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  • DOI: https://doi.org/10.1007/BF03167201

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