Sendov's conjecture holds for every degree $n\ge 10^{200000}$

Abstract

Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial $f$ of degree $n\ge 2$ has all its zeros in the closed unit disk $z\in \mathbb{C}:|z|\le 1$, then, for every zero ${\lambda }_0$ of $f$, there exists a critical point $\zeta$ of $f$ such that $|\zeta -{\lambda }_0|\le 1$. The conjecture was previously known to hold for polynomials of degree $n\le 8$, as well as in several special higher-degree cases. In 2022, using compactness methods, balayage, and the argument principle, Tao~\cite{Tao22} proved that there exists an absolute constant $n_0$ such that Sendov's conjecture holds for all $n\ge n_0$. However, Tao's argument does not provide an explicit admissible value of $n_0$. In the present paper, we make Tao's result effective and prove that one may take $$n_0=10^{200000}.$$

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