mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...

and critical points of a

polynomial function In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ...

of a complex variable. It is named after Blagovest Sendov. The

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

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 , ''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 In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometry, geometric relation between the root of a function, roots of a polynomial and the roots of its derivative . The set of roots of a real or complex polynomial ...

says that all of the critical points lie within the

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

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 does not have the said property across all its ordered instances, but will after some instances have ...

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History

The conjecture was first proposed by Blagovest Sendov in 1959; he described the conjecture to his colleague Nikola Obreshkov. In 1967 the conjecture was misattributed to Ljubomir Iliev by Walter Hayman. 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 The ''Journal of Mathematical Analysis and Applications'' is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960 by Richard Bellman, as part of a series of new ...

232 (1999), no. 2, 272–292. proved the conjecture for ''n'' < 9 in 1999. Terence Tao 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

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Complex analysis Conjectures Unsolved problems in mathematics