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In mathematics, '''Sendov's conjecture''', sometimes also called '''Ilieff's conjecture''', concerns the relationship between the locations of [[zero of a function|roots]] and [[critical point (mathematics)|critical point]]s of a [[polynomial function]] of a [[complex variable]]. It is named after [[Blagovest Sendov]].

The conjecture states that for a polynomial

: <math> f(z) = (z - r_1)\cdots (z-r_n),\qquad (n\ge 2) </math>

with all roots ''r''<sub>1</sub>, ..., ''r''<sub>''n''</sub> 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 not been proved for ''n'' > 8.

== References ==

* G. Schmeisser, "The Conjectures of Sendov and [[Stephen Smale|Smale]]," ''Approximation Theory: A Volume Dedicated to Blagovest Sendov'' (B. Bojoanov, ed.), Sofia: DARBA, 2002 pp. 353–369.

== External links ==

* [http://demonstrations.wolfram.com/SendovsConjecture/ Sendov's Conjecture] by Bruce Torrence with contributions from Paul Abbott at [[The Wolfram Demonstrations Project]]

[[Category:Complex analysis]]
[[Category:Conjectures]]

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