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

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.