Egerv\'ary's theorems for harmonic trinomials

Abstract

In this manuscript, we study the arrangements of the roots in the complex plane for the lacunary harmonic polynomials called harmonic trinomials. We provide necessary and sufficient conditions so that two general harmonic trinomials have the same set of roots up to a rotation around the origin in the complex plane, a reflection over the real axis, or a composition of the previous both transformations. This extends the results of J. Egerv\'ary 1930 for the setting of trinomials to the setting of harmonic trinomials.