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

Let P be the family of complex-valued polynomials of the form p(z) = (z 1)(z r1)(z r2)2 with |r1| = |r2| = 1. The Gauss–Lucas theorem guarantees that the critical points of p 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 :|z 3 4| < 1 4} and the interior of 2x4 3x3 + x + 4x2y2 3xy2 + 2y4 = 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.

geometry of polynomials, critical points, Gauss–Lucas theorem
Mathematical Subject Classification 2010
Primary: 30C15
Received: 21 February 2017
Revised: 5 June 2017
Accepted: 13 June 2017
Published: 20 October 2017

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