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.
