Geometry of a Family of Complex Polynomials

Abstract

Let P_a 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 P_a 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 P_a 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 P_a.