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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∈ Kθp satisfy P(f,g,h)=0, in which P is a homogeneous polynomial and the associated projective plane curve is smooth. Then dimspan{f,g,h}≤1 if either deg P≥ 4 and the boundary spectrum of θ has logarithmic capacity zero, or 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.
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BibTeX
@misc{garcia2026fermats,
title = {{Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation}},
author = {Stephan Ramon Garcia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17180},
eprint = {2610.17180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17180},
}
RIS
TY - UNPB
TI - Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation
AU - Stephan Ramon Garcia
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17180
UR - https://arxiv.org/abs/2610.17180
AB - We study algebraic relations among functions in star-invariant subspaces. Let θ be an inner function and suppose that f,g,h∈ 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 dimspan{f,g,h}≤1 if either deg P≥ 4 and the boundary spectrum of θ has logarithmic capacity zero, or 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.
ER -