= Solution
A <plasma wave resonance> has $n^2\to\infty$. At finite coefficients of the <magnetized cold-plasma dielectric tensor>, its generic condition is the vanishing of the coefficient of $n^4$:
$$
\boxed{S\sin^2\theta+P\cos^2\theta=0,\qquad \tan^2\theta=-P/S.}
$$
A <plasma wave cutoff> has $n^2=0$, so the constant term of the <dispersion relation> vanishes:
$$
\boxed{PRL=0,\quad\text{that is }P=0,\ R=0,\ \text{or }L=0.}
$$
Both statements are branch conditions: check the undivided <dispersion relation> at special angles and simultaneous zero factors. In particular, the parallel <plasma oscillation> at $P=0$ is a longitudinal branch with arbitrary $k$ in the cold limit, while the transverse parallel branches have <plasma wave cutoffs> at $R=0$ or $L=0$. Poles of the <magnetized cold-plasma dielectric tensor> can give additional limiting <plasma wave resonances>, such as the parallel <cold-plasma right-handed wave> at $\omega=\Omega_e$; these are not obtained by assuming finite $S,P,g$.
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