Year: 2026
Type: preprint
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 $λ_0$ of $f$, there exists a critical point $ζ$ of $f$ such that $|ζ-λ_0| \leq 1$. 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 $$ n_0 = 10^{200000}. $$
Source: arXiv (Cornell University)
Author: Zhang, Teng
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213585460