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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the zeros of a polynomial and its derivatives


Author: Piotr Pawlowski
Journal: Trans. Amer. Math. Soc. 350 (1998), 4461-4472
MSC (1991): Primary 30C15
MathSciNet review: 1473453
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Abstract: If $p(z)$ is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of $p'(z)$ lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of $p(z)$ to a nearest zero of $\displaystyle{p'(z)}$? We obtain bounds for this distance depending on degree. We also show that this distance is equal to $\frac{1}{3}$ for polynomials of degree 3 and polynomials with real zeros.


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Additional Information

Piotr Pawlowski
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
Address at time of publication: Summit Systems, Inc., 22 Cortland St., New York, New York 10007
Email: piotr-pawlowski@summithq.com

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02291-0
PII: S 0002-9947(98)02291-0
Keywords: Polynomials, location of zeros
Received by editor(s): June 27, 1996
Article copyright: © Copyright 1998 American Mathematical Society