# Verification of the algebraic zero curve theorem

Alper Ferudun, Mercury Software GmbH. Version 1.0, 5 October 2026.

The manuscript proves the full irreducible-curve version of Stensones's
2010 AIM question in every degree. Every irreducible affine algebraic
curve in C^2 avoiding zero on which a nonzero homogeneous psh polynomial
vanishes is exactly a nonzero homogeneous holomorphic level. If
reducible curves are admitted, the manuscript gives a counterexample
even without pluriharmonic terms. This is not merely the quadratic case.

The originating researcher audited finite Laurent pole orders on the
compact normalization, convergence of the rescaled Levi matrices,
rotational averaging, the scalar determinant obstruction, the survival
and elimination of harmonic terms, bounded removable ends, polarization,
nonvanishing of the dilation polynomial, and exact irreducible fiber
equality. No mathematical gap is currently identified. This is self-audit,
not independent peer review or a proof-assistant certificate.

The supporting Python/Sympy script passes 39 exact regression checks:
the general Levi determinant, transverse jets through order twelve,
harmonic Laurent terms including negative exponents, the reducible
example, several meromorphic monomial-fiber parametrizations, and
countermodels showing why positivity and the global algebraic hypothesis
matter. These tests do not replace the written analytic proof.

The final standalone LaTeX source passed the desktop editor's compiler.
The five-page PDF was exported with the already-installed Tectonic
without TeX warnings. All five final page renders were visually checked,
including the name-only author line, affiliation/contact footnote, math
symbols and bibliography. AI assistance is disclosed in an ordinary
paragraph of the manuscript.

The upstream quadratic result and elementary reducible obstruction are
credited. Bharali and Stensones and Simon supply related factorization
results with additional foliation hypotheses; the proof here does not
assume those hypotheses. The literature check was bounded and did not
establish novelty or absolute priority. Publication in a repository does
not constitute independent mathematical endorsement.