On the Parameter Spaces of Harmonic Trinomial Equations

TL;DR

This paper investigates the geometric structure of the parameter space of harmonic trinomial equations, focusing on curves called trochoids that arise from root multiplicity and modulus conditions.

Contribution

It introduces a detailed geometric analysis of parameter space curves for harmonic trinomials, extending classical theorems to this context.

Findings

Identification of specific geometric curves in parameter space

Characterization of curves associated with root multiplicities

Analysis of the properties of trochoids in harmonic trinomial equations

Abstract

We analyze the parameter space of harmonic trinomial equations of the form $z^{n+m}+b\overline{z}^m+c$, where $n,m\in\mathbb{Z}^+$ are coprime and $b,c\in\mathbb{C}$. Using versions of the Bohl and Egerv\'ary theorems for harmonic trinomials, we describe the geometric curves in the parameter space that arise when considering a simple root or a multiple root, or when two distinct roots have the same modulus. In particular, we study the geometric properties of these curves, called trochoids.