Geometry of polynomials with three roots (Q721089)

The author studies polynomials of the form \[ P(k,n,m)=\{P: \mathbb{C}\mapsto \mathbb{C} \;\mid \; p(z)=(z-1)^k(z-r_1)^m(z-r_2)^n , |r_1|=|r_2|=1,\} \] where \(k,m,n\in \mathbb{N}, n\neq m.\) By the Gauss-Lucas Theorem, the critical points of such a polynomials are within the unit disk. This paper characterizes the relationship between the zeros and the critical points of polynomials in \(P(k,m,n)\). In particular, it is shown that the unit disk contains two regions in which critical points cannot occur, and each \(c\) inside the unit disk and strictly outside of the above two regions is the critical point of exactly two polynomials in \(P(k,n,m).\)