A note on the connectedness locus of the families of polynomials P(c)(z)=z(n) - cz(n-j)

An Acad Bras Cienc. 2012 Mar;84(1):5-8. doi: 10.1590/s0001-37652012000100002.

Abstract

Let j be a positive integer. For each integer n > j we consider the connectedness locus M(n) of the family of polynomials P(c)(z)=z(n) - cz(n-j), where c is a complex parameter. We prove that lim n→∞ M(n) = D in the Hausdorff topology, where D is the unitary closed disk {c;|c|≤1}.