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Title: Generalization of a Conjecture in the Geometry of Polynomials
Authors: Sendov, Bl.
Keywords: Geometry of Polynomials
Gauss-Lucas Theorem
Zeros of Polynomials
Critical Points
Ilieff-Sendov Conjecture
Issue Date: 2002
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 28, No 4, (2002), 283p-304p
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.
ISSN: 1310-6600
Appears in Collections:Volume 28 Number 4

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