Vol. 31, No. 2, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 330: 1
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
On Ilyeff’s conjecture

A. Meir and Ambikeshwar Sharma

Vol. 31 (1969), No. 2, 459–467

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
Received: 20 December 1968
Published: 1 November 1969
A. Meir
Ambikeshwar Sharma