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How to prove that a non-quadratic complex polynomial $P(z)=a+bz+cz^2+dz^3$ with $a,b,c,d\in \Bbb C$ has a closed curve on $\Bbb C$ which is not the image of a circle under a Möbius transformation?

For a non-quadratic complex polynomial $P(z)=a+bz+cz^2+dz^3$ with $a,b,c,d\in \Bbb C$, how to show that there exists an automorphism $T$ on the Riemann sphere such that $T(P(z))=w$ has a closed curve on $\Bbb C$ which is not the image of a circle under a Möbius transformation?

A:

The set of all quadratic polynomials without real roots is open and dense in the space of all complex polynomials. It’s not known whether the set of all cubic polynomials without real roots is, but I would bet it is. By the way, $P(z)=az^2+bz^3+cz^4+dz^5$ has $3$ real roots (or you can count them explicitly, e.g., use Vieta’s formulas).
Edit: it turns out that the set of all cubic polynomials without real roots is not open (Example A). Also, the set of all real cubic polynomials without real roots has codimension $1$, and therefore it’s non-smooth.

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Andrew Tabler is a senior fellow and director of the Russia and Eurasia program at the Washington Institute for Near East Policy. He served on the National Security Council Staff at the White House and held positions at the State Department, National Security Council and Defense Department. His articles and reviews have appeared in numerous publications, including the Wall Street Journal, Washington Post, National Interest, Defense News, San Francisco Chronicle, and World Politics Review. He is a graduate of Yale University.

Simone Dennison is a graduate of the University of Virginia, and the author of Flag Politics: How the American Flag Shapes the Foreign Policy Politics of the United States. Her books include Under the Big Top: Anatomy of an American Confection in the New Diplomacy, and The Peacock Fanfare: Music, Diplomacy, and the Myth of American Exceptionalism. She currently serves on the editorial board of the journal Diplomatic History.

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