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We derive a formula for the number of pre-images under a non-degenerateharmonic mapping $f$, using the argument principle. This formula reveals aconnection between the pre-images and the caustics. Our results allow to deducethe 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 $k = n, n+1, \ldots, n^2$ thereexists a harmonic polynomial of degree $n$ with $k$ zeros.

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