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On the Sendov Conjecture for Sixth Degree Polynomials

Johnny E. Brown
Proceedings of the American Mathematical Society
Vol. 113, No. 4 (Dec., 1991), pp. 939-946
DOI: 10.2307/2048768
Stable URL: http://www.jstor.org/stable/2048768
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Sendov Conjecture for Sixth Degree Polynomials
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Abstract

The Sendov conjecture asserts that if $p(z) = \prod^n_{k = 1}(z - z_k)$ is a polynomial with zeros |zk| ≤ 1, then each disk |z - zk| ≤ 1, (1 ≤ k ≤ 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 |zj| ≤ λ6 = 0.626997... then |z - zj| ≤ 1 contains a zero of p'(z). (The conjecture for n = 6 would be proved if λ6 = 1.) It is shown that λ6 can be improved to λ6 = 63/64 = 0.984375.

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