Approximate factorization of polynomials over Z

  • Authors:

  • Tateaki Sasaki;Yasutaka Ookura

  • Affiliations:

  • University of Tsukuba, Tsukuba-Shi, Japan;University of Tsukuba, Tsukuba-Shi, Japan

  • Venue:
  • Proceedings of the 2009 conference on Symbolic numeric computation
  • Year:
  • 2009

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Abstract

We propose three algorithms for approximate factorization of univariate polynomials over Z; the first one uses sums of powers of roots (SPR method), the second one utilizes factor-differentiated polynomials (FD method), and the third one is a robust but slow method. The SPR method is applicable to monic polynomials well but it is almost useless for non-monic polynomials unless their leading coefficients are sufficiently small. The FD method is applicable to both monic and non-monic polynomials, but it also becomes useless if both the leading and the tail coefficients increase. The third one is applicable to any polynomial factorizable approximately over Z, but it is slow. We discuss two types of polynomials which are ill-conditioned for rootfinding, Wilkinson-type polynomials and polynomials with close roots. Furthermore, we consider briefly approximate factorization of multivariate polynomials over Z.