On the Shapes of Rational Lemniscates

Christopher J. Bishop; Alexandre Eremenko; Kirill Lazebnik

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arXiv · 2407.14610

On the Shapes of Rational Lemniscates

Christopher J. Bishop·Alexandre Eremenko·Kirill Lazebnik

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Abstract

A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge's theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.

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Christopher J. Bishop, Alexandre Eremenko, Kirill Lazebnik. 2024-07-19. On the Shapes of Rational Lemniscates. https://arxiv.org/abs/2407.14610

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