Roots multiplicity without companion matrices

We show a method for constructing a polynomial interpolating roots' multiplicities of another polynomial, that does not use companion matrices. This leads to a modification to Guersenzvaig--Szechtman square-free decomposition algorithm that is more efficient both in theory and in practice.

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References

  • [1] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).
  • [2] Natalio H. Guersenzvaig and Fernando Szechtman. Roots multiplicity and square-free factorization of polynomials using companion matrices. Linear Algebra Appl., 436(9):3160–3164, 2012.
  • [3] Ellis Horowitz. Algorithms for symbolic integration of rational functions. PhD thesis, University of Wisconsin, 1969.
  • [4] François Le Gall.

    Powers of tensors and fast matrix multiplication.

    In ISSAC 2014—Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pages 296–303. ACM, New York, 2014.
  • [5] David Rea Musser. Algorithms for polynomial factorization. PhD thesis, University of Wisconsin, 1971.
  • [6] Robert G. Tobey. Algorithms for antidifferentiation of rational functions. PhD thesis, Harvard, 1967.
  • [7] David Y.Y. Yun. On square-free decomposition algorithms. In Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation, SYMSAC ’76, pages 26–35, New York, NY, USA, 1976. ACM.
  • [8] D. V. Zhdanovich. Exponent of complexity of matrix multiplication. Fundam. Prikl. Mat., 17(2):107–166, 2011/12.