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On the Sendov conjecture for sixth degree polynomials


Author: Johnny E. Brown
Journal: Proc. Amer. Math. Soc. 113 (1991), 939-946
MSC: Primary 30C15
DOI: https://doi.org/10.1090/S0002-9939-1991-1081693-X
MathSciNet review: 1081693
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Abstract: The Sendov conjecture asserts that if $ p(z) = \prod _{k = 1}^n(z - {z_k})$ is a polynomial with zeros $ \left\vert {{z_k}} \right\vert \leq 1$, then each disk $ \left\vert {z - {z_k}} \right\vert \leq 1,(1 \leq k \leq n)$ contains a zero of $ p'(z)$. This conjecture has been verified in general only for polynomials of degree $ n = 2,3,4,5$. If $ p(z)$ is an extremal polynomial for this conjecture when $ n = 6$, it is known that if a zero $ \left\vert {{z_j}} \right\vert \leq {\lambda _6} = 0.626997 \ldots $ then $ \left\vert {z - {z_j}} \right\vert \leq 1$ contains a zero of $ p'(z)$. (The conjecture for $ n = 6$ would be proved if $ {\lambda _6} = 1$.) It is shown that $ {\lambda _6}$ can be improved to $ {\lambda _6} = 63/64 = 0.984375$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1081693-X
Article copyright: © Copyright 1991 American Mathematical Society

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