Vol. 31, No. 2, 1969

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On Ilyeff’s conjecture

A. Meir and Ambikeshwar Sharma

Vol. 31 (1969), No. 2, 459–467
Abstract

An apparently easy problem due to Ilyeff states: If all zeros z1,z2,,zn of a complex polynomial P(z) lie in |z|1 then there is always a zero of P(z) in each of the disks |z zj|1,j = 1,,n. If true, the conjecture is best possible as one can see from the example P(z) = zn 1. In full generality the conjectured result was proved only for polynomials of degree 4. In this paper the conjecture is proved for quintics and extensions of earlier results are obtained for zeros of higher derivatives of polynomials having multiple roots.

Mathematical Subject Classification
Primary: 30.11
Milestones
Received: 20 December 1968
Published: 1 November 1969
Authors
A. Meir
Ambikeshwar Sharma