Sendov conjecture for high degree polynomials

Details Sendov conjecture for high degree polynomials Sendov conjecture for high degree polynomials

2011-06-21

arxiv.org/abs/1106.4126v2

Mathematics

Complex Variables

14 pages, 5 figures

Scientific paper

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture.

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