Sendov's conjecture

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

f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)

with all roots r₁, ..., 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 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]

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^[3] to Ljubomir Iliev by Walter Hayman.^[4] 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^[5] proved the conjecture for n < 9 in 1999. Terence Tao proved the conjecture for sufficiently large n in 2020.

References

^ Terence Tao (2020). "Sendov's conjecture for sufficiently high degree polynomials". arXiv:2012.04125 [math.CV].
^ Terence Tao (9 December 2020). "Sendov's conjecture for sufficiently high degree polynomials". What's new.
^ Marden, Morris. Conjectures on the Critical Points of a Polynomial. The American Mathematical Monthly 90 (1983), no. 4, 267-276.
^ Problem 4.5, W. K. Hayman, Research Problems in Function Theory. Althlone Press, London, 1967.
^ 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.

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.