TY - JOUR
AU - Felton, Camille Elizabeth
AU - Frayer, Christopher
PY - 2019/10/27
TI - Critical Points of a Family of Complex-Valued Polynomials
JF - Minnesota Journal of Undergraduate Mathematics; Vol 5 No 1 (2019): Minnesota Journal of Undergraduate Mathematics
KW -
N2 - 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 $P(k,m,n)$ .
UR - https://mjum.math.umn.edu/index.php/mjum/article/view/117