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We give sufficient conditions under which a polyanalytic polynomial of degree $n$ has (i) at least one zero, and (ii) finitely many zeros. In the latter case, we prove that the number of zeros is bounded by $n^2$. We then show that for all $k \in \{0,\dots, n^2, \infty\}$ there exists a polyanalytic polynomial of degree $n$ with exactly $k$ distinct zeros. Moreover, we generalize the Lagrange and Cauchy bounds from analytic to polyanalytic polynomials and obtain inclusion disks for the zeros. Finally, we construct a harmonic and thus polyanalytic polynomial of degree $n$ with $n$ nonzero coefficients and the maximum number of $n^2$ zeros., Comment: 22 pages, 4 figures

We present a continuation method to compute all zeros of a harmonic mapping $f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solution of $f(z) = \eta$ with $|\eta|$ sufficiently large and then track all solutions as $\eta$ tends to $0$ to finally obtain all zeros of $f$. Using theoretical results on harmonic mappings we analyze where and how the number of solutions of $f(z) = \eta$ changes and incorporate this into the method. We prove that our method is guaranteed to compute all zeros, as long as none of them is singular. In our numerical example the method always terminates with the correct number of zeros, is very fast compared to general purpose root finders and is highly accurate in terms of the residual. An easy-to-use MATLAB implementation is freely available online.

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., Comment: 27 pages, 7 figures

We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton's method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of $f = h + \bar{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z) = \eta$ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. We provide a Matlab implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction., Comment: 26 pages, 10 figures. Improved visualization of the dynamics of the harmonic Newton map. Some minor further improvements

Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions $f(z) = \frac{p(z)}{q(z)} - \overline{z}$, which depend on both $\mathrm{deg}(p)$ and $\mathrm{deg}(q)$. Furthermore, we prove that any function that attains one of these upper bounds is regular., Comment: Accepted for publication in CMFT

We study the effect of constant shifts on the zeros of rational harmomic functions $f(z) = r(z) - \conj{z}$. In particular, we characterize how shifting through the caustics of $f$ changes the number of zeros and their respective orientations. This also yields insight into the nature of the singular zeros of $f$. Our results have applications in gravitational lensing theory, where certain such functions $f$ represent gravitational point-mass lenses, and a constant shift can be interpreted as the position of the light source of the lens., Comment: 26 pages, 9 figures