Generalization of a Conjecture in the Geometry of Polynomials

Sendov, Bl.. "Generalization of a Conjecture in the Geometry of Polynomials." Serdica Mathematical Journal 28.4 (2002): 283-304. <http://eudml.org/doc/11565>.

@article{Sendov2002,
abstract = {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.},
author = {Sendov, Bl.},
journal = {Serdica Mathematical Journal},
keywords = {Geometry of Polynomials; Gauss-Lucas Theorem; Zeros of Polynomials; Critical Points; Ilieff-Sendov Conjecture; geometry of polynomials; Gauss-Lucas theorem; Ilieff-Sendov conjecture},
language = {eng},
number = {4},
pages = {283-304},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Generalization of a Conjecture in the Geometry of Polynomials},
url = {http://eudml.org/doc/11565},
volume = {28},
year = {2002},
}

TY - JOUR
AU - Sendov, Bl.
TI - Generalization of a Conjecture in the Geometry of Polynomials
JO - Serdica Mathematical Journal
PY - 2002
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 28
IS - 4
SP - 283
EP - 304
AB - 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.
LA - eng
KW - Geometry of Polynomials; Gauss-Lucas Theorem; Zeros of Polynomials; Critical Points; Ilieff-Sendov Conjecture; geometry of polynomials; Gauss-Lucas theorem; Ilieff-Sendov conjecture
UR - http://eudml.org/doc/11565
ER -