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.