On the Parameter Spaces of Harmonic Trinomial Equations

TL;DR

The paper studies parameter spaces for harmonic trinomial equations of the form $h(z)=a z^{n+m}+b\overline{z}^m+c$ with coprime $n,m$, extending classical analytic-trinomial results to the harmonic setting. It leverages specialized Bohl and Egerváry theorems to describe the loci in $(b,c)$-space where a root of a given modulus occurs, showing these loci are trochoids, and that two roots share a modulus precisely when $b$ lies on a union of rays. A detailed analysis yields the topological structure of the root-modulus sets $U_j$, including their nonconnectedness and the bounds on the total number of roots up to $n+3m$. The work highlights the role of trochoid geometry and the Jacobian in understanding multiplicity and discriminant-like behavior, offering a harmonic-extension of prior analytic results without relying on Amoeba theory. It provides a robust framework for characterizing how parameter changes influence root distributions in harmonic trinomial families.

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 Egerváry 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.

Figures (6)

• Figure 1: Roots $z_1, z_2,\ldots,z_{10}$ of the harmonic trinomial $h(z)=z^4-5\overline{z}\,^3+2$. And the circles $C_1$, $C_2$ and $C_3$ with radio $v = \tfrac{1}{2}, 2, \text{ and } 6$ respectively.
• Figure 2: Parametrization of $h_1$, $h_2$ y $h_3$ with a root of norm $v=1$ to trochoids with parameters $R$, $r$ y $d$.
• Figure 3: Parametrization of $h_4$, $h_5$ y $h_6$ with a root of $v=1$ to trochoids with parameters $R$, $r$ y $d$.
• Figure 4: Trochoid associated to $h=z^{2}-2\overline{z}+1$ taking a root with modulus $\sqrt{5}$. The dotted lines represent the even rays ($\mathcal{R}^{\mathrm{even}}$) and the dashed lines represent the odd rays ($\mathcal{R}^{\mathrm{odd}}$). We observe that $b=-2\in\mathcal{R}$.
• Figure 6: Trochoid associated to $h = z^{5} + 6\,\overline{z}^{3} + 1$, taking a complex root with modulus $v \approx 0.55589$. The dotted lines represent the even rays ($\mathcal{R}^{\mathrm{even}}$) and the dashed lines represent the odd rays ($\mathcal{R}^{\mathrm{odd}}$). We observe that $b = 6 \notin \mathcal{R}^{\mathrm{odd}}$.

Theorems & Definitions (50)

• Definition 1.1
• Theorem 1.2: Bohl for harmonic trinomials, barrera2022number
• Example 1.3
• Theorem 1.4: Bohl for harmonic trinomials, barrera2022number, BARRERA2024128213
• Example 1.5
• Definition 1.6
• Theorem 1.7: Egerváry for harmonic trinomials, barrera2024egervary
• Example 1.8
• Definition 1.9
• Definition 1.10