In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after 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 r1, ..., rn inside the closed 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 says that all of the critical points lie within the 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 by Tao.[1][2

In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after 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 r1, ..., rn inside the closed 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 says that all of the critical points lie within the 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 by Tao.[1][2