View source for Sendov's conjecture

{{short description|Conjecture about the roots of polynomials}}
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|polynomial function]] of a complex variable.  It is named after [[Biography:Blagovest Sendov|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>,&nbsp;...,&nbsp;''r''<sub>''n''</sub> inside the closed [[Unit disk|unit disk]] |''z''|&nbsp;≤&nbsp;1, each of the ''n'' roots is at a distance no more than&nbsp;1 from at least one critical point.

The [[Gauss–Lucas theorem]] says that all of the critical points lie within the [[Convex hull|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&nbsp;''n''&nbsp;<&nbsp;9 by Brown-Xiang and for&nbsp;''n'' [[Eventually (mathematics)|sufficiently large]] by [[Biography:Terence Tao|Tao]].<ref name="Tao2020arxiv">{{cite arXiv |eprint=2012.04125 |author=Terence Tao |title=Sendov's conjecture for sufficiently high degree polynomials|year=2020 |class=math.CV }}</ref><ref>{{Cite web|author=Terence Tao|website=What's new|url=https://terrytao.wordpress.com/2020/12/08/sendovs-conjecture-for-sufficiently-high-degree-polynomials/|title=Sendov's conjecture for sufficiently high degree polynomials |date=9 December 2020}}</ref>

==History==

The conjecture was first proposed by [[Biography:Blagovest Sendov|Blagovest Sendov]] in 1959; he described the conjecture to his colleague Nikola Obreshkov. In 1967 the conjecture was misattributed<ref>Marden, Morris. Conjectures on the Critical Points of a Polynomial. [[The American Mathematical Monthly]] 90 (1983), no. 4, 267-276.</ref> to Ljubomir Iliev by [[Biography:Walter Hayman|Walter Hayman]].<ref>Problem 4.5, W. K. Hayman, Research Problems in Function Theory. Althlone Press, London, 1967.</ref> In 1969 Meir and Sharma proved the conjecture for polynomials with ''n'' < 6. In 1991 Brown proved the conjecture for ''n'' < 7. Borcea extended the proof to ''n'' < 8 in 1996. Brown and Xiang<ref>Brown, Johnny E.; Xiang, Guangping Proof of the Sendov conjecture for polynomials of degree at most eight. [[Journal of Mathematical Analysis and Applications]] 232 (1999), no. 2, 272–292.</ref> proved the conjecture for ''n'' < 9 in 1999. [[Biography:Terence Tao|Terence Tao]] proved the conjecture for sufficiently large ''n'' in 2020.

== References ==
{{Reflist}}

* G. Schmeisser, "The Conjectures of Sendov and [[Biography:Stephen Smale|Smale]]," ''Approximation Theory: A Volume Dedicated to Blagovest Sendov'' (B. Bojoanov, ed.), Sofia: DARBA, 2002 pp.&nbsp;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]]
[[Category:Unsolved problems in mathematics]]

{{Sourceattribution|Sendov's conjecture}}