Sète, OlivierZur, Jan2022-08-012022-08-012021-06-212737-0690https://depositonce.tu-berlin.de/handle/11303/17287http://dx.doi.org/10.14279/depositonce-16067We 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.en518 Numerische Analysisharmonic mappingspre-imagescausticsargument principlevalencezeros of harmonic polynomialsArticle2737-114X