The Sendov conjecture asserts that if a polynomial polynomial degree roots in the closed
unit disk , then every root distance
1 of a critical point of Lucas's root theorem , the critical
points , which are the roots of the derivative ,
also lie in the closed unit disk .
For a quadratic polynomial critical point
is triangle inequality gives
where antipodal points
on the unit circle . The constant 1 is sharp for every
polynomial degree critical
point is the origin , and all roots
are roots of unity at distance
1 from it.
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.
See also Borcea Variance Conjecture ,
Critical Point ,
Lucas's
Root Theorem ,
Phelps-Rodriguez Conjecture ,
Polynomial ,
Unit Disk
Portions of this entry contributed by Bruce
Torrence
Explore with Wolfram|Alpha
References Marden, M. "Conjectures on the Critical Points of a Polynomial." Amer. Math. Monthly 90 , 267-276, 1983. Prasolov,
V. V. Polynomials. https://doi.org/10.1007/978-3-642-03980-5 . 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/ . Trott,
M. "Polynomials in Inequalities." §1.2.3 in The
Mathematica GuideBook for Symbolics. https://doi.org/10.1007/0-387-28815-5 .
https://www.mathematicaguidebooks.org/ . Referenced
on Wolfram|Alpha Sendov Conjecture
Cite this as:
Weisstein, Eric W. , with contributions by Bruce Torrence . "Sendov Conjecture." From MathWorld https://mathworld.wolfram.com/SendovConjecture.html
Subject classifications More... Less...
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.
with 
, the unique critical point
is 
.
The triangle inequality gives
,
2. Equality holds exactly when 
and 
are antipodal points
on the unit circle. The constant 1 is sharp for every
polynomial degree 
. For 
, the only critical
point is the origin, and all roots
are roots of unity at distance
1 from it.
