On the zeros of a polynomial and its derivatives
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- by Piotr Pawlowski PDF
- Trans. Amer. Math. Soc. 350 (1998), 4461-4472 Request permission
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 $p’(z)$? 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.References
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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
- Received by editor(s): June 27, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 4461-4472
- MSC (1991): Primary 30C15
- DOI: https://doi.org/10.1090/S0002-9947-98-02291-0
- MathSciNet review: 1473453