Geometry of a Class of Generalized Cubic Polynomials

Abstract

This paper studies a class of generalized complex cubic polynomials of the form p(z)=(z-1)(z-r_1)^k(z-r_2)^k where r_1 and r_2 lie on the unit circle and k is a natural number.   We completely characterize where the nontrivial critical points of p can lie, and to what extent they determine the polynomial.     The main results include (1) a nontrivial critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a `desert' in the unit disk in which critical points cannot occur.