# Generalization of a Conjecture in the Geometry of Polynomials

Serdica Mathematical Journal (2002)

- Volume: 28, Issue: 4, page 283-304
- ISSN: 1310-6600

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topSendov, 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 -

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