Functions of a complex variable Geometric function theory Conjecture

Sendov's conjecture

Builds on Root of a polynomial Critical point Convex hull

Statement

Let $n \ge 2$ and let $p$ be a polynomial of degree $n$ with complex coefficients whose roots all lie in the closed unit disc $\set{z \in \C : \abs{z} \le 1}$ . Then for each root $a$ of $p$ there is a root $b$ of the derivative $p'$ with $\abs{a - b} \le 1$ .

Remarks

The roots of $p'$ are the critical points of $p$ . The Gauss–Lucas theorem says that the critical points lie in the convex hull of the roots; this only gives the distance bound $2$ . The bound $1$ would be sharp: for $p(z) = z^n - 1$ and the root $a = 1$ , the only critical point is $0$ , at distance exactly $1$ . The conjecture has been proved for polynomials of degree at most $8$ , by Brown and Xiang. Terence Tao proved it for all polynomials of sufficiently large degree $n$ . The general case is open.

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Tag	`101H2VSX2R`
URI	`m2://101H2VSX2R`
Kind	conjecture
Writing	W0 stub
MSC	30C15, 30C10
Status	This copy of the site has no build data. Run `m2 status 101H2VSX2R`.
About	Root of a polynomial, Critical point
Uses	Root of a polynomial, Critical point, Convex hull
Same as	Wikidata Q17037028 Wikipedia Sendov's conjecture
Changed	2 November 2026

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