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

TL;DR

This paper develops a formula using the argument principle to determine the number and location of pre-images under harmonic mappings, linking them to caustics and enabling geometric deductions and polynomial zero constructions.

Contribution

It introduces a novel formula for pre-image counting under harmonic maps and connects pre-images to caustics, with applications to harmonic polynomial zeros.

Findings

Derived a formula for pre-image count using the argument principle

Connected pre-images to caustics and provided geometric location methods

Proved existence of harmonic polynomials with a range of zeros

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.