Minnesota Journal of Undergraduate Mathematics

Critical Points of a Family of Complex-Valued Polynomials

Camille Elizabeth Felton

University of Wisconsin Platteville

Christopher Frayer

University of Wisconsin-Platteville

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Abstract

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 $P(k,m,n)$.