A tale of two circles: geometry of a class of quartic polynomials

Abstract

Let $\mathcal{P}$ be the family of complex-valued polynomials of the form $p\left(z\right)=\left(z-1\right)\left(z-r_1\right){\left(z-r_2\right)}^2$ with $\left|r_1\right|=\left|r_2\right|=1$. The Gauss–Lucas theorem guarantees that the critical points of $p\in \mathcal{P}$ will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk $\left\{z\in ℂ:\left|z-\frac{3}{4}\right|<\frac{1}{4}\right\}$ and the interior of $2x^4-3x^3+x+4x^2{y}^2-3x{y}^2+2y^4=0$, in which critical points of $p$ cannot occur. Furthermore, each $c$ inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in $\mathcal{P}$.