Open Access
November 2017 Geometry of Polynomials with Three Roots
Christopher Frayer
Missouri J. Math. Sci. 29(2): 161-175 (November 2017). DOI: 10.35834/mjms/1513306828


Given a complex-valued polynomial of the form $p(z) = (z-1)^k(z-r_1)^m(z-r_2)^n$ with $|r_1|=|r_2|=1$; $k,m,n \in \mathbb{N}$ and $m \neq n$, where are the critical points? The Gauss-Lucas Theorem guarantees that the critical points of such a polynomial will lie within the unit disk. This paper further explores the location and structure of these critical points. Surprisingly, the unit disk contains two ‘desert’ regions in which critical points cannot occur, and each $c$ inside the unit disk and outside of the desert regions is the critical point of exactly two such polynomials.


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Christopher Frayer. "Geometry of Polynomials with Three Roots." Missouri J. Math. Sci. 29 (2) 161 - 175, November 2017.


Published: November 2017
First available in Project Euclid: 15 December 2017

zbMATH: 06905062
MathSciNet: MR3737294
Digital Object Identifier: 10.35834/mjms/1513306828

Primary: 30C15

Keywords: critical points , Gauss-Lucas theorem , geometry of polynomials

Rights: Copyright © 2017 Central Missouri State University, Department of Mathematics and Computer Science

Vol.29 • No. 2 • November 2017
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