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

Abstract

We study algebraic relations among functions in star-invariant subspaces. Let $θ$ be an inner function and suppose that $f,g,h\in K_θ^p$ satisfy $P(f,g,h)=0$, in which $P$ is a homogeneous polynomial and the associated projective plane curve is smooth. Then $\dim\operatorname{span}\{f,g,h\}\leq1$ if either $\operatorname{deg} P\geq 4$ and the boundary spectrum of $θ$ has logarithmic capacity zero, or $\operatorname{deg} P = 3$ 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.