Unimodular Zeros of Selmer's Harmonic Trinomials

Let $n>k$ be positive integers and let $\epsilon ,\delta \in \{-1,1\}$. In this note, we classify the unimodular zeros of the harmonic trinomial $H_{\epsilon ,\delta }:\mathbb{C}\to \mathbb{C}$ given by $$H_{\epsilon ,\delta }(z)=z^n+\epsilon {\bar{z}}^k+\delta ,$$ where $\bar{z}$ 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 $n_0:=n\mathrm{/}g$ and $k_0:=k\mathrm{/}g$. For each choice of $\epsilon ,\delta \in \{-1,1\}$, we give a necessary and sufficient congruence involving $n_0$ and $k_0$ to guarantee that $H_{\epsilon ,\delta }$ has unimodular zeros. More precisely, if the corresponding congruence holds, then $H_{\epsilon ,\delta }$ 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.