Zeros of a one-parameter family of harmonic trinomials

Abstract

It is well known that complex harmonic polynomials of degree n n may have more than n n zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.