Location of the Zeros of Certain Complex-Valued Harmonic Polynomials

TL;DR

This paper explores the zero locations of complex-valued harmonic polynomials, extending known results for analytic polynomials to a broader class and providing bounds for zeros of harmonic trinomials.

Contribution

It generalizes zero inclusion regions to complex harmonic polynomials and bounds zeros of harmonic trinomials, advancing understanding in this less-studied area.

Findings

Zeros of harmonic trinomials are contained in specific annular regions.

General zero inclusion regions for harmonic polynomials are established.

Bounds for zeros of particular harmonic polynomial families are provided.

Abstract

Finding an approximate region containing all the zeros of analytic polynomials is a well-studied problem. But the numb er of the zeros and regions containing all the zeros of complex-valued harmonic polynomials is relatively a fresh research area. It is well known that all the zeros of analytic trinomials are enclosed in some annular sectors that take into account the magnitude of the coefficients. Following Kennedy and Dehmer, we provide the zero inclusion regions of all the zeros of complex-valued harmonic polynomials in general, and in particular, we bound all the zeros of some families of harmonic trinomials in a certain annular region.