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arXiv · 1908.08759

Number and location of pre-images under harmonic mappings in the plane

Olivier Sète·Jan Zur
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Abstract

We derive a formula for the number of pre-images under a non-degenerate harmonic mapping $f$, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under $f$ geometrically for every non-caustic point. We approximately locate the pre-images of points near the caustics. Moreover, we apply our results to prove that for every $k = n, n+1, \ldots, n^2$ there exists a harmonic polynomial of degree $n$ with $k$ zeros.

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Olivier Sète, Jan Zur. 2019-08-23. Number and location of pre-images under harmonic mappings in the plane. https://doi.org/10.5186/aasfm.2021.4614

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