Sendov conjecture for high degree polynomials

Jérôme Dégot

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arXiv · 1106.4126

Sendov conjecture for high degree polynomials

Jérôme Dégot

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Abstract

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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BibTeX RIS

Jérôme Dégot. 2011-11-15. Sendov conjecture for high degree polynomials. https://arxiv.org/abs/1106.4126

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