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

Frayer, Christopher; Gauthier, Landon

doi:10.2140/involve.2018.11.489

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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

DOI: 10.2140/involve.2018.11.489

Abstract

Let $P$ be the family of complex-valued polynomials of the form $p (z) = (z - 1) (z - r_{1}) {(z - r_{2})}^{2}$ with $| r_{1} | = | r_{2} | = 1$ . The Gauss–Lucas theorem guarantees that the critical points of $p \in P$ 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 ${z \in ℂ : | z - \frac{3}{4} | < \frac{1}{4}}$ and the interior of $2 x^{4} - 3 x^{3} + x + 4 x^{2} y^{2} - 3 x y^{2} + 2 y^{4} = 0$ , 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 $P$ .

Keywords

geometry of polynomials, critical points, Gauss–Lucas theorem

Mathematical Subject Classification 2010

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

Vol. 11, No. 3, 2018