On the number of roots for harmonic trinomials

TL;DR

This paper investigates the roots of harmonic trinomials, establishing their root count, and proves the Fundamental Theorem of Algebra and Wilmshurst conjecture for these polynomials using the Bohl method.

Contribution

It provides the first comprehensive root counting for harmonic trinomials and confirms key conjectures in the field.

Findings

Root count for harmonic trinomials established

Fundamental Theorem of Algebra proven for harmonic trinomials

Wilmshurst conjecture validated for harmonic trinomials

Abstract

In this manuscript we study the counting problem for harmonic trinomials of the form $a\zeta^n+b\overline{\zeta}^m+c$, where $n,m\in \mathbb{N}$, $n>m$, and $a$, $b$ and $c$ are non-zero complex numbers. As a consequence, we obtain the Fundamental Theorem of Algebra and the Wilmshurst conjecture for harmonic trinomials. The proof of the counting problem relies on the Bohl method introduced in Bohl (1908).