Sendov’s conjecture

Prove that for every integer n ≥ 2, the optimal constant C_sendov(n) equals 1; that is, for any complex polynomial f of degree n whose zeros lie in the unit disk, every zero ζ of f has a critical point ξ of f′ within distance 1 (|ξ − ζ| ≤ 1).

Background

Sendov’s conjecture (also known as Ilyeff’s conjecture) is a classical problem in complex analysis on the location of critical points relative to zeros of polynomials with constrained root geometry. Partial progress includes proofs for small degrees and asymptotic regimes, but the general case remains unresolved.

Exact determination of C_sendov(n) has implications for potential theory, polynomial geometry, and extremal configuration analysis.