Geodesic
Generalization of a Conjecture in the Geometry of Polynomials
Serdica Mathematical Journal, Tome 28 (2002) no. 4, pp. 283-304.

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In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.
Keywords: Geometry of Polynomials, Gauss-Lucas Theorem, Zeros of Polynomials, Critical Points, Ilieff-Sendov Conjecture 
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Sendov, Bl. Generalization of a Conjecture in the Geometry of Polynomials. Serdica Mathematical Journal, Tome 28 (2002) no. 4, pp. 283-304. http://geodesic.mathdoc.fr/item/SMJ2_2002_28_4_a2/