Sendov’s conjecture for sufficiently-high-degree polynomials

\emph{Sendov's conjecture} asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all of its zeroes in closed unit disk $\{ z: |z| \leq 1 \}$, then for each such zero $\lambda_0$ there is a zero of the derivative $f'$ in the closed unit disk $\{ z: |z-\lambda_0| \leq 1 \}$. 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 $n \geq n_0$. For $\lambda_0$ away from the origin and the unit circle we can appeal to the prior work of Degot 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.