On the Parameter Spaces of Harmonic Trinomial Equations

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.