On the number of roots 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).