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, . . . , n 2 there exists a harmonic polynomial of degree n with k zeros.

The transport of images method: computing all zeros of harmonic mappings by continuation

“…The theoretical foundation of the transport of images method uses results of Lyzzaik [21], Neumann [23] and our previous results in [29,28]. We prove that our method is guaranteed to compute all zeros of a non-degenerate harmonic mapping f .…”

“…We briefly present relevant results for the computation of all zeros of harmonic mappings, following the lines of [28] and [21].…”

“…with analytic h, g, but are themselves non-analytic in general. Several results on the number of zeros for special classes of harmonic mappings are known in the literature, e.g., for harmonic polynomials, i.e., f (z) = p(z) + q(z), where p and q are analytic polynomials [35,15,9,16], for rational harmonic mappings of the form f (z) = r(z)−z, where r is a rational function [13,5,19,20,27,26,17,18,28], or certain transcendental harmonic mappings [8,4,12]. While the above publications have a theoretical focus, we are here interested in the numerical computation of the zeros of f .…”

“…The problem with such an approach is, however, that in general the number of zeros of harmonic mappings is not known a priori. In fact, even for harmonic polynomials f (z) = p(z) − z with deg(p) = n ≥ 2, the number of zeros can vary between n and 3n − 2, and all these different numbers of zeros are attained by some harmonic polynomial; see [28,Thm. 5.4].…”

“…This effect depends on the caustics of f (curves of the critical values). More precisely, the number of solutions of f (z) = η changes by ±2 when η changes from one side to the other of a single caustic arc [28]. For every step from η k to η k+1 , we construct a (minimal) set of points S k+1 , such that all solutions of f (z) = η k+1 are obtained by applying Newton's method with initial points from S k+1 .…”