#### Vol. 11, No. 3, 2018

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

### Christopher Frayer and Landon Gauthier

Vol. 11 (2018), No. 3, 489–500
##### Abstract

Let $\mathsc{P}$ 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 $|{r}_{1}|=|{r}_{2}|=1$. The Gauss–Lucas theorem guarantees that the critical points of $p\in \mathsc{P}$ 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 $\left\{z\in ℂ:|z-\frac{3}{4}|<\frac{1}{4}\right\}$ and the interior of $2{x}^{4}-3{x}^{3}+x+4{x}^{2}{y}^{2}-3x{y}^{2}+2{y}^{4}=0$, 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 $\mathsc{P}$.

##### Keywords
geometry of polynomials, critical points, Gauss–Lucas theorem
Primary: 30C15
##### Milestones
Received: 21 February 2017
Revised: 5 June 2017
Accepted: 13 June 2017
Published: 20 October 2017

Communicated by Michael Dorff
##### Authors
 Christopher Frayer Department of Mathematics University of Wisconsin Platteville, WI United States Landon Gauthier University of Wisconsin Platteville, WI United States