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 In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...

of a

complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...

. It is named after

Blagovest Sendov Blagovest Hristov Sendov ( bg, Благовест Сендов; 8 February 1932 – 19 January 2020) was a Bulgarian mathematician, diplomat and politician. Early life and education Sendov was born in Asenovgrad, Bulgaria. Career Academi ...

. The conjecture states that for a polynomial :

f(z) = (z - r_1)\cdots (z-r_n),\qquad (n\ge 2)

with all roots ''r''₁, ..., ''r''_''n'' inside the closed

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...

, ''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 In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have pa ...

by Tao.

History

The conjecture was first proposed by

in 1959; he described the conjecture to his colleague

Nikola Obreshkov Nikola Dimitrov Obreshkov ( bg, Никола Димитров Обрешков) (March 6, 1896 in VarnaAugust 11, 1963 in Sofia) was a prominent Bulgarian mathematician, working in complex analysis Complex analysis, traditionally known as ...

. In 1967 the conjecture was misattributed to Ljubomir Iliev by

Walter Hayman Walter Kurt Hayman FRS (6 January 1926 – 1 January 2020) was a British mathematician known for contributions to complex analysis. He was a professor at Imperial College London. Life and work Hayman was born in Cologne, Germany, the son ...

. 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 XiangBrown, 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. proved the conjecture for ''n'' < 9 in 1999.

Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...

proved the conjecture for sufficiently large ''n'' in 2020.

References

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

External links

Sendov's Conjecture
by Bruce Torrence with contributions from Paul Abbott at

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

Complex analysis Conjectures Unsolved problems in mathematics