Extending a result of Khavinson and Świa̧tek (2003) we show that the rational harmonic
function $\overline {r (z)}-z $, where $ r (z) $ is a rational function of degree $ n> 1$, has no …

Complex-valued analytic functions have many very nice properties that are not necessarily
possessed by real-valued functions. For example, we say a complex-valued function is …

We demonstrate counterexamples to Wilmshurst's conjecture on the valence of harmonic
polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for …

The fundamental theorem of algebra (FTA) tells us that every complex polynomial of degree
n has precisely n complex roots. The first published proofs (including those of J. d'Alembert …

We give a topological characterization of rational maps with disconnected Julia sets. Our
results extend Thurston's characterization of postcritically finite rational maps. In place of …

Let f. z/D pz/= qz/be a rational map with p and q relatively prime polynomials. The degree d
D deg. f/of f is defined to be the maximum of the degrees of p and q. In the following we will …

Motivated by Wilmshurst's conjecture and more recent work of W. Li and A. Wei [17], we
determine asymptotics for the number of zeros of random harmonic polynomials sampled …

We show that there is a one-to-one correspondence between conjugacy classes of critically
fixed anti-rational maps and equivalence classes of certain plane graphs. We furthermore …

We prove the existence of complex polynomials $ p (z) $ of degree $ n $ and $ q (z) $ of
degree $ m< n $ such that the harmonic polynomial $ p (z)+\overline {q (z)} $ has at least …

We study the distribution of complex zeros of Gaussian harmonic polynomials with
independent complex coefficients. The expected number of zeros is evaluated by applying a …