Geometry of a Family of Complex Polynomials

  • Christopher Frayer University of Wisconsin-Platteville
  • Jamison Wallace University of Wisconisn-Platteville
Keywords: geometry of polynomials, critical points, Gauss-Lucas Theorem


Let Pa be the family of complex-valued polynomials of the form p(z)=(z-a)(z-r)(z-s) with a in [0,1] and r and s on the unit circle.   The Gauss-Lucas Theorem implies that the critical points of a polynomial in Pa lie in the unit disk.  This paper characterizes the location and structure of these critical points.  We show that the unit disk contains an open circular disk in which critical points of polynomials in Pa do not occur.  Furthermore, almost every c inside the unit disk and outside of the desert region is the critical point of a unique polynomial in Pa.

Author Biographies

Christopher Frayer, University of Wisconsin-Platteville

Chris earned a Bachelor of Science from Grand Valley State University in 2003, a PhD from the University of Kentucky in 2008, and just finished his 10th year of teaching mathematics at UW-Platteville. In his spare time, he enjoys spending time with his wife and two daughters, being outdoors, and reading.

Jamison Wallace, University of Wisconisn-Platteville

Jamison is a mathematics major in his final year at UW-Platteville and has been involved in three undergraduate research groups (two in chemistry and one in mathematics). He is also a fencing coach and vice president of the UW-Platteville fencing club.

How to Cite
Frayer, C., & Wallace, J. (2019). Geometry of a Family of Complex Polynomials. The PUMP Journal of Undergraduate Research, 2, 95-106. Retrieved from