Zeros of a Family of Complex-Valued Harmonic Trinomials

It is known that complex-valued harmonic polynomials of degree n can have more than n zeros. In the 2020 paper “Zeros of a One-Parameter Family of Harmonic Trinomials” by Brilleslyper et al., the authors consider a one-parameter family of complex-valued harmonic polynomials and determine, for different values of the real parameter, the number of zeros. We consider the same family but allow the parameter to be complex. As in the previous paper, our proof relies on the Argument Principle for Harmonic Functions and again requires us to find the winding number about the origin of a hypocycloid. The geometry is more complicated in this case, however. This additional complexity is reflected in the main theorem, which shows that the number of transitions in the number of zeros and the nature of these transitions depends on the argument of the complex parameter.