Zeros of a Two-parameter Family of Harmonic Quadrinomials

TL;DR

This paper analyzes the zeros of a specific two-parameter family of harmonic quadrinomials, identifying their zero distribution, inclusion regions, and a dividing curve for sense-preserving and reversing regions, using classical and modern mathematical tools.

Contribution

It provides a detailed characterization of zeros and zero regions for this family of harmonic quadrinomials, extending previous work with new geometric and algebraic insights.

Findings

Number of zeros determined

Zero inclusion regions identified

Curve separating sense-preserving and reversing regions established

Abstract

In this paper, we determine the numb er of zeros and the zero inclusion regions of a two-parameter family of harmonic quadrinomials. We also determine a curve that separates sense-preserving and sense-reversing regions for these families of quadrinomials. Our work makes practical and effective use of the work of Wilmshurst, Khavinson, Dehmer, and also Bezouts theorem in the plane.