Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation

We study algebraic relations among functions in star-invariant subspaces. Let $\theta$ be an inner function and suppose that $f,g,h\in K_{\theta }^p$ satisfy $P(f,g,h)=0$, in which $P$ is a homogeneous polynomial and the associated projective plane curve is smooth. Then $\dim \mathrm{span}\{f,g,h\}\le 1$ if either $\mathrm{deg}P\ge 4$ and the boundary spectrum of $\theta$ has logarithmic capacity zero, or $\mathrm{deg}P=3$ and the boundary spectrum contains at most one point. This settles new cases of a problem of Dyakonov, subsumes the previously known results, and recovers the classical fact that polynomial parametrizations of a smooth projective plane curve of degree at least three are projectively constant.