An Explicit High-Degree Threshold for Sendov’s Conjecture

Jeff Pickhardt

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An Explicit High-Degree Threshold for Sendov’s Conjecture

Abstract

Let p be a polynomial of degree n ≥2 whose zeros lie in the closed unit disk. Sendov’s conjecture asserts that every zero aof plies within distance 1 of a zero of p′. Tao proved that the conjecture holds for all sufficiently large degrees, but the compactness and unique-continuation arguments in his proof did not provide an explicit threshold. We give an effective version. Specifically, we prove that Sendov’s conjecture holds whenever n ≥ exp(10^126). The principal new ingredient is a quantitative treatment of the near-unit-circle regime. A local Cauchy-transform estimate is propagated directly by harmonic measure through an eccentric annulus.