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

Published electronically:
June 3, 2002

MathSciNet review:
1929045

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

**[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****[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*, American Mathematical Society Colloquium Publications, Vol. 34, American Mathematical Society, New York, N. Y., 1950. MR**0037350**

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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:
http://dx.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