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

Sète, Olivier; Zur, Jan

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, ... , n^2 there exists a harmonic polynomial of degree n with k zeros.
Published in: Annales Fennici Mathematici, 10.5186/aasfm.2021.4614, The Finnish Mathematical Society