Sendov’s conjecture for sufficiently-high-degree polynomials

Sendov’s conjecture asserts that if a complex polynomial $f$ of degree $n \geqslant 2$ has all of its zeros in closed unit disk $\lbrace z: \vert z \vert \leqslant 1 \rbrace$, then for each such zero $\lambda_0$ there is a zero of the derivative $f^\prime$ in the closed unit disk $\lbrace z: \vert z-\lambda_0 \vert \leqslant 1 \rbrace$. This conjecture is known for $n \lt 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov’s conjecture holds for $n \geqslant n_0$. For $\lambda_0$ away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for $\lambda_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $\lambda_0$ is extremely close to the unit circle); and for $\lambda_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.

2010 Mathematics Subject Classification

30C15, 31A05

Full Text (PDF format)

Received 7 December 2020

Accepted 31 May 2022

Published 21 February 2023