# Sendov's conjecture

By [xWhiteOuroboros](https://paragraph.com/@xwhiteouroboros) · 2024-01-27

---

In mathematics, **Sendov's conjecture**, sometimes also called **Ilieff's conjecture**, concerns the relationship between the locations of [roots](https://en.wikipedia.org/wiki/Zero_of_a_function) and [critical points](https://en.wikipedia.org/wiki/Critical_point_\(mathematics\)) of a [polynomial function](https://en.wikipedia.org/wiki/Polynomial_function) of a [complex variable](https://en.wikipedia.org/wiki/Complex_variable). It is named after [Blagovest Sendov](https://en.wikipedia.org/wiki/Blagovest_Sendov).

The conjecture states that for a polynomial

$${\\displaystyle f(z)=(z-r\_{1})\\cdots (z-r\_{n}),\\qquad (n\\geq 2)}$$

with all roots _r_1, ..., _rn_ inside the closed [unit disk](https://en.wikipedia.org/wiki/Unit_disk) |_z_| ≤ 1, each of the _n_ roots is at a distance no more than 1 from at least one critical point.

The [Gauss–Lucas theorem](https://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem) says that all of the critical points lie within the [convex hull](https://en.wikipedia.org/wiki/Convex_hull) of the roots. It follows that the critical points must be within the unit disk, since the roots are.

The conjecture has been proven for _n_ < 9 by Brown-Xiang and for _n_ [sufficiently large](https://en.wikipedia.org/wiki/Eventually_\(mathematics\)) by [Tao](https://en.wikipedia.org/wiki/Terence_Tao).[\[1\]](https://en.wikipedia.org/wiki/Sendov%27s_conjecture#cite_note-Tao2020arxiv-1)[\[2](https://en.wikipedia.org/wiki/Sendov%27s_conjecture#cite_note-2)

---

*Originally published on [xWhiteOuroboros](https://paragraph.com/@xwhiteouroboros/sendov-s-conjecture)*