On the Sendov conjecture for sixth degree polynomials

Brown, Johnny E.

doi:10.1090/S0002-9939-1991-1081693-X

On the Sendov conjecture for sixth degree polynomials
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by Johnny E. Brown PDF

Proc. Amer. Math. Soc. 113 (1991), 939-946 Request permission

Abstract:

The Sendov conjecture asserts that if $p(z) = \prod _{k = 1}^n(z - {z_k})$ is a polynomial with zeros $\left | {{z_k}} \right | \leq 1$, then each disk $\left | {z - {z_k}} \right | \leq 1,(1 \leq k \leq n)$ 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 $\left | {{z_j}} \right | \leq {\lambda _6} = 0.626997 \ldots$ then $\left | {z - {z_j}} \right | \leq 1$ 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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On Ilieff’s conjecture

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