Sendov Conjecture

The Sendov conjecture, proposed by Blagovest Sendov circa 1958, that for a polynomial with and each root located inside the closed unit disk in the complex plane, it must be the case that every closed disk of radius 1 centered at a root will contain a critical point of . Since the Lucas-Gauss theorem implies that the critical points (i.e., the roots of the derivative) of must themselves lie in the unit disk, it seems completely implausible that the conjecture could be false. Yet at present it has not been proved even for polynomials with real coefficients, nor for any polynomials whose degree exceeds eight.