The conjecture was established for polynomial degrees at most 8 and, by Tao (2022), for all sufficiently large polynomial
degrees. Tao (2026) described, checked, and reorganized an AI-generated proof
obtained by L. Mazur with a Lean formalization, showing that it also proves
the stronger Phelps-Rodriguez conjecture,
and supplied a separate Lean formalization of his exposition.
Marden, M. "Conjectures on the Critical Points of a Polynomial." Amer. Math. Monthly90, 267-276, 1983.Rahman,
Q. I. and Schmeisser, G. Analytic Theory of Polynomials. Oxford, England:
Oxford University Press, 2002.Schmeisser, G. "The Conjectures of
Sendov and Smale." In Approximation Theory: A Volume Dedicated to Blagovest
Sendov (Ed. B. Bojoanov). Sofia, Bulgaria: DARBA, pp. 353-369, 2002.Sendov,
Bl. "Generalization of a Conjecture in the Geometry of Polynomials." Serdica
Math. J.28, 283-304, 2002.Tao, T. "Sendov's Conjecture
for Sufficiently-High-Degree Polynomials." Acta Math.229, 347-392,
2022. https://doi.org/10.4310/ACTA.2022.v229.n2.a3.Tao,
T. "A Digestion of the Proof of Sendov's Conjecture." Aug. 12, 2026.
https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/.
has polynomial degree 
and all its roots in the closed
unit disk, then every root 
is within distance 1 of a critical
point of 
.
It originated as an unpublished 1958 proposal by Blagovest Sendov (Sendov 2002),
who communicated it to Marden in 1962 (Marden 1983). By Lucas's
root theorem, the critical points, which are
the roots of the derivative,
also lie in the closed unit disk.
