Unimodular Zeros of Selmer's Harmonic Trinomials

Abstract

Let $n>k$ be positive integers and let $\varepsilon,δ\in\{-1,1\}$. In this note, we classify the unimodular zeros of the harmonic trinomial $H_{\varepsilon,δ}:\mathbb{C}\to \mathbb{C}$ given by \[ H_{\varepsilon,δ}(z)=z^n+\varepsilon\overline z^{\,k}+δ, \] where $\overline{z}$ is the complex conjugate of $z\in \mathbb{C}$. Let $g:=\gcd(n,k)$ be the greatest common divisor of $n$ and $k$, and set $n_0:=n/g$ and $k_0:=k/g$. For each choice of $\varepsilon,δ\in \{-1,1\}$, we give a necessary and sufficient congruence involving $n_0$ and $k_0$ to guarantee that $H_{\varepsilon,δ}$ has unimodular zeros. More precisely, if the corresponding congruence holds, then $H_{\varepsilon,δ}$ has exactly $2g$ unimodular zeros, which are given explicitly; otherwise, it has no unimodular zeros. The proof uses a reduction to a trinomial equation and elementary divisibility arguments.