On Ilyeff’s conjecture

Abstract

An apparently easy problem due to Ilyeff states: If all zeros z L , z z , , z n of a complex polynomial P(z) lie in | z | ^ 1 then there is always a zero of P r (z) in each of the disks I z -z 3 ? I ^ 1, j -1, , n. If true, the conjecture is best possible as one can see from the example P(z) = z n -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.