Sendov's conjecture for sufficiently high degree polynomials

Terence Tao

Sendov's conjecture for sufficiently high degree polynomials

TL;DR

The paper proves that Sendov's conjecture holds for all sufficiently large degrees by a compactness-contradiction framework that analyzes asymptotic behavior of polynomial zeros. It builds a probabilistic-analytic model with random zeros

and

, deriving limiting objects

and

via potential theory, Balayage, and Stieltjes transforms. It handles two endpoint regimes,

and

, proving that both lead to consistent limiting behavior that cannot violate the conjecture, and then tackles the origin and unit-circle delicate regimes with a mix of argument-principle arguments and Taylor expansions to derive final contradictions. While the method is non-constructive and yields an existence of

without an explicit value, it lays groundwork toward a decidability result for the conjecture at high degrees and clarifies the asymptotic structure of zeroes and critical points of high-degree polynomials in the unit disk context.

Abstract

Sendov's conjecture asserts that if a complex polynomial

of degree

has all of its zeroes in closed unit disk

, then for each such zero

there is a zero of the derivative

in the closed unit disk

. This conjecture is known for

, but only partial results are available for higher

. We show that there exists a constant

such that Sendov's conjecture holds for

. For

away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for

near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when

is extremely close to the unit circle); and for

near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.