Geodesic
Number and location of pre-images under harmonic mappings in the plane
Olivier Sète1 ; Jan Zur1
1TU Berlin, Department of Mathematics, MA 3-3
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 225-247
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  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.
Keywords: harmonic mappings, pre-images, caustics, argument principle, valence, zeros of harmonic polynomials

Olivier Sète  1   ; Jan Zur  1

1 TU Berlin, Department of Mathematics, MA 3-3 
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Olivier Sète; Jan Zur. Number and location of pre-images under harmonic mappings in the plane. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 225-247. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a12/