Sendov's conjecture for sufficiently high degree polynomials

Abstract

Sendov's conjecture asserts that if a complex polynomial $f$ of degree $n\ge 2$ has all of its zeroes in closed unit disk $\{z:|z|\le 1\}$, then for each such zero ${\lambda }_0$ there is a zero of the derivative $f^{\prime }$ in the closed unit disk $\{z:|z-{\lambda }_0|\le 1\}$. This conjecture is known for $n<9$, 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 $n\ge n_0$. 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.

38 pages, 5 figures. To appear, Acta Mathematica. Referee comments incorporated

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