---
tag: 101H2VSX2R
kind: conjecture
name: Sendov's conjecture
msc: [30C15, 30C10]
uses: [019MA1VTR2, 0YTCMFB4HC, 0FQSDC2EQ2]
about: [019MA1VTR2, 0YTCMFB4HC]
sources:
  - url: https://en.wikipedia.org/wiki/Sendov%27s_conjecture
same_as:
  wikidata: Q17037028
  wikipedia: Sendov's conjecture
history:
  posed_by: "Blagovest Sendov"
  first_stated: "1959"
---

## Statement

Let $n \ge 2$ and let $p$ be a polynomial of degree $n$ with complex coefficients whose [[roots|019MA1VTR2]] 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|0YTCMFB4HC]] of $p$.
The Gauss–Lucas theorem says that the critical points lie in the [[convex hull|0FQSDC2EQ2]] 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.

## Explanations

### L1 human

Place the roots of a polynomial inside the unit disc.
The derivative of the polynomial has its own roots, which tend to sit among the roots of the polynomial.
The conjecture says that every root of the polynomial has a root of the derivative within distance $1$.
It is simple to state, yet it has resisted proof for decades.