Geometry of a Family of Complex Polynomials

Abstract

Let Pa be the family of complex-valued polynomials of the form p(z)=(z-a)(z-r)(z-s) with a in [0,1] and r and s on the unit circle. The Gauss-Lucas Theorem implies that the critical points of a polynomial in Pa lie in the unit disk. This paper characterizes the location and structure of these critical points. We show that the unit disk contains an open circular disk in which critical points of polynomials in Pa do not occur. Furthermore, almost every c inside the unit disk and outside of the desert region is the critical point of a unique polynomial in Pa.