%A Felton, Camille Elizabeth
%A Frayer, Christopher
%D 2019
%T Critical Points of a Family of Complex-Valued Polynomials
%K
%X For $k,m,n \in \mathbb{N}$ , let $P(k,m,n)$ be the family of complex-valued polynomials of the form $p(z) = z^k(z-r_1)^m(z-r_2)^n$ with $|r_1|=|r_2|=1$ . The Gauss-Lucas Theorem guarantees that the critical points of $p \in P(k,m,n)$ will lie in the unit disk. This paper further explores the location and structure of these critical points. When $m=n$ , the unit disk contains a \emph {desert region}, $ \{ z \in \mathbb{C} : |z| < \frac{k}{k+2m} \},$ in which critical points do not occur, and a critical point almost always determines a polynomial uniquely. When $m \neq n$ , the unit disk contains two desert regions, and each $c$ is the critical point of at most two polynomials in $P(k,m,n)$ .
%U https://mjum.math.umn.edu/index.php/mjum/article/view/117
%J Minnesota Journal of Undergraduate Mathematics
%0 Journal Article
%V 5
%N 1
%@ 2378-5810
%8 2019-10-27