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The Geometry of Cubic Polynomials

Christopher Frayer, Miyeon Kwon, Christopher Schafhauser and James A. Swenson
Mathematics Magazine
Vol. 87, No. 2 (April 2014), pp. 113-124
DOI: 10.4169/math.mag.87.2.113
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Page Count: 12
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Item Type
The Geometry of Cubic Polynomials
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Summary We study the critical points of a complex cubic polynomial, normalized to have the form \documentclass{article} \pagestyle{empty}\begin{document} $p(z) = (z-1)(z-r_1)(z-r_2)$ \end{document} with \documentclass{article} \pagestyle{empty}\begin{document} $|r_1|=1=|r_2|$ \end{document} . If Tγ denotes the circle of diameter passing through 1 and 1 − γ, then there are α, β ∈ [0, 2] such that one critical point of p lies on Tα and the other on Tβ. We show that Tβ is the inversion of Tα over T1, from which many geometric consequences can be drawn. For example, (1) a critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a “desert” in the unit disk, the open disk \documentclass{article} \pagestyle{empty}\begin{document} $\{ z \in \mathbb{C} : |z- \frac23| < \frac13 \}$ \end{document} , in which critical points cannot occur.

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