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 $n \geq 2$ has all its zeros in the closed unit disk ${z \in \mathbb{C} : |z| \leq 1}$, then, for every zero $\lambda_0$ of $f$, there exists a critical point $\zeta$ of $f$ such that $|\zeta-\lambda_0| \leq 1$. The conjecture was previously known to hold for polynomials of degree $n \leq 8$, 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 $n \geq n_0$. 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 $$ n_0 = 10^{200000}. $$