10.30495/maca.2021.1938758.1028

On the zeros and critical points of a polynomial

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  • Mohammad Ibrahim Mir1,
  • Irfan Ahmad Wani1,
  • Ishfaq Nazir1,
  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India

Published in Issue 2022-01-01

How to Cite

On the zeros and critical points of a polynomial. (2022). Mathematical Analysis and Its Contemporary Applications, 4(1). https://doi.org/10.30495/maca.2021.1938758.1028

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Abstract

Let P(z) = a0 + a1z + ... + an-1z^{n-1} + z^n be a polynomial of degree n. The Gauss-Lucas Theorem asserts that the zeros of the derivative P'(z) = a1 + ... + (n - 1)an-1z^{n-2} + nz^{n-1}, lie in the convex hull of the zeros of P(z). Given a zero of P(z) or P'(z), A. Aziz [1], determined regions which contain at least one zero of P(z) or P'(z) respectively. In this paper, we give simple proofs and improved version of various results proved in [1], concerning the zeros of a polynomial and its derivative.

Keywords

  • polynomial,
  • zeros,
  • critical points,
  • half plane,
  • circular region
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References

  1. A. Aziz, On the zeros of a Polynomials and its derivative, Bull. Aust. Math. Soc., 31(4)(1985), 245-255.
  2. J. Brown and G. Xiang, Proof of the Sendov conjecture for the polynomial of degree atmost eight, J. Math. Anal. Appl., 232(1999), 272-292.
  3. A. W. Goodman, Q. I. Rahman and J. S. Ratti, On the zeros of a polynomial and it's derivative, Proc. Amer. Math. Soc., 21(1969), 273-274.
  4. Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, Oxford University Press, 2002.