On the Ilieff-Sendov conjecture

Abstract

The well-known Ilieff-Sendov conjecture asserts that for any polynomial p(z) =XT k=l{z-zk) wit h ^\ ^i <1, each of thedisks. W\<: 1 (1< k< n) must contain a critical point of p. This conjecture is proved for polynomials of arbitrary degree n with at most four distinct zeros. This extends a result of Saff and Twomey.

On zero-free regions for the derivative of a polynomial

“…Actually conjecture was due to a Bulgarian mathematician B. Sendov. In connection with this conjecture Brown [4] posed the following problem. …”

The Sendov Conjecture for Polynomials With at Most Seven Distinct Zeros

“…It has also been proved that those roots of p lying sufEciently dose to the Unit circle satisfy an even stronger condition than the one stated in Sendov's conjecture ( [13], [19]). Various other special cases have been dealt with (see [3], [9], and [18] for references), the latest to date being that of polynomials with at most six different zeros, and more generally polynomials in P"(i/(n)), where v{n) is an increasing and unbounded function of n (Theorem 2.4 below) (cf. [2]; see also [6]).…”

Continuous independence and the Ilieff-Sendov conjecture

“…Numerous attempts to verify this conjecture have led to over 30 papers, but have met with limited success (for references, see [Ma2,B]). …”