2020/12/08 by Terence Tao, Tao, Terence · 1 citation
Mathematics · #30C15 #31A05 #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2012.04125
openalex publication_date 2020/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Sendov's conjecture asserts that if a complex polynomial f of degree n ≥ 2 has all of its zeroes in closed unit disk \ z: |z| ≤ 1 \, then for each such zero λ0 there is a zero of the derivative f' in the closed unit disk \ z: |z-λ0| ≤ 1 \. This conjecture is known for n < 9, but only partial results are available for higher n. We show that there exists a constant n0 such that Sendov's conjecture holds for n ≥ n0. For λ0 away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for λ0 near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when λ0 is extremely close to the unit circle); and for λ0 near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.