Egerv\'ary's theorems for harmonic trinomials

TL;DR

This paper extends Egerváry's classical theorems to harmonic trinomials, characterizing root arrangements and symmetries in the complex plane, thus broadening understanding of harmonic polynomial root structures.

Contribution

It provides necessary and sufficient conditions for harmonic trinomials to have roots related by rotations, reflections, or both, generalizing Egerváry's results from algebraic to harmonic polynomials.

Findings

Characterization of root arrangements for harmonic trinomials

Conditions for roots to be related by symmetries

Extension of Egerváry's theorems to harmonic polynomials

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.