Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the location of critical points of polynomials


Authors: Branko Curgus and Vania Mascioni
Journal: Proc. Amer. Math. Soc. 131 (2003), 253-264
MSC (2000): Primary 30C15; Secondary 26C10
DOI: https://doi.org/10.1090/S0002-9939-02-06534-6
Published electronically: June 3, 2002
MathSciNet review: 1929045
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a polynomial $p$ of degree $n \geq 2$ and with at least two distinct roots let $Z(p) = \{z : p(z) = 0\}$. For a fixed root $\alpha \in Z(p)$ we define the quantities $\omega(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p)\setminus \{\alpha\} \bigr\}$and $\tau(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p')\setminus \{\alpha\} \bigr\}$. We also define $\omega(p)$ and $\tau(p)$ to be the corresponding minima of $\omega(p,\alpha)$ and $\tau(p,\alpha)$ as $\alpha$ runs over $Z(p)$. Our main results show that the ratios $\tau(p,\alpha)/\omega(p,\alpha)$ and $\tau(p)/\omega(p)$ are bounded above and below by constants that only depend on the degree of $p$. In particular, we prove that $(1/n)\omega(p)\leq\tau(p)\leq\bigl(1/2\sin(\pi/n)\bigr)\omega(p)$, for any polynomial of degree $n$.


References [Enhancements On Off] (What's this?)

  • [1] Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
  • [2] Peter Henrici, Applied and computational complex analysis. Vol. 1, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Power series—integration—conformal mapping—location of zeros; Reprint of the 1974 original; A Wiley-Interscience Publication. MR 1008928
  • [3] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
  • [4] Maurice Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992. Translated from the French by Catherine Mignotte. MR 1140923
  • [5] A. M. Ostrowski, On the moduli of zeros of derivatives of polynomials, J. Reine Angew. Math. 230 (1968), 40–50. MR 0225973, https://doi.org/10.1515/crll.1968.230.40
  • [6] J. L. Walsh, On the location of the roots of the derivative of a polynomial, C. R. Congr. Internat. des Mathématiciens, pp. 339-342, Strasbourg, 1920.
  • [7] J. L. Walsh, The location of critical points of analytic and harmonic functions, Amer. Math. Soc. Colloq. Publ., Vol. 34, 1950. MR 12:249d

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30C15, 26C10

Retrieve articles in all journals with MSC (2000): 30C15, 26C10


Additional Information

Branko Curgus
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
Email: curgus@cc.wwu.edu

Vania Mascioni
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
Email: masciov@cc.wwu.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06534-6
Keywords: Roots of polynomials, critical points of polynomials, separation of roots
Received by editor(s): July 10, 2001
Received by editor(s) in revised form: September 4, 2001
Published electronically: June 3, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society