Sendov conjecture for high degree polynomials

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 $P^{\mathrm{\prime }}$. 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 $\mathrm{\mid }a\mathrm{\mid }$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P^{\mathrm{\prime }}$) of a polynomial which would contradict Sendov conjecture.