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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maximal polynomials and the Ilieff-Sendov conjecture

Author: Michael J. Miller
Journal: Trans. Amer. Math. Soc. 321 (1990), 285-303
MSC: Primary 30C15; Secondary 26C10, 30C10
MathSciNet review: 965744
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Abstract: In this paper, we consider those complex polynomials which have all their roots in the unit disk, one fixed root, and all the roots of their first derivatives as far as possible from a fixed point. We conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point, and as many of its own roots as possible on the unit circle. We prove a part of this conjecture, and use it to define an algorithm for constructing some of these polynomials. With this algorithm, we investigate the 1962 conjecture of Sendov and the 1969 conjecture of Goodman, Rahman and Ratti and (independently) Schmeisser, obtaining counterexamples of degrees $6$, $8$, $10$, and $12$ for the latter.

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Keywords: Ilieff, Sendov, geometry of polynomials, roots of polynomials, maximal polynomials
Article copyright: © Copyright 1990 American Mathematical Society