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

We derive a formula for the number of pre-images under non-degenerate harmonic mapping $f$, using argument principle. This reveals connection between and caustics. Our results allow to deduce $f$ geometrically every non-caustic point. approximately locate points near Moreover, we apply our prove that $k = n, n+1, \ldots, n^2$ there exists polynomial degree $n$ with $k$ zeros.