Critical Points of a Family of Complex-Valued Polynomials

  • Camille Elizabeth Felton University of Wisconsin Platteville
  • Christopher Frayer University of Wisconsin-Platteville


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 $P(k,m,n)$.
How to Cite
FELTON, Camille Elizabeth; FRAYER, Christopher. Critical Points of a Family of Complex-Valued Polynomials. Minnesota Journal of Undergraduate Mathematics, [S.l.], v. 5, n. 1, oct. 2019. ISSN 2378-5810. Available at: <>. Date accessed: 25 sep. 2020.