We study the critical points of complex-valued polynomials of the form  with  and  The Gauss-Lucas theorem guarantees that the critical points of such a polynomial will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, there is a ‘desert’, the open disk  in which the critical points cannot occur. Furthermore, a critical point of such a polynomial almost always determines the polynomial uniquely.

geometry of polynomials, critical points, Marden’s theorem, Gauss-Lucas theorem.