Algebraic Zero Curves of Homogeneous Plurisubharmonic Polynomials
Manuscript 5 October 2026 · Online 5 October 2026
Abstract
Let \(p\) be a nonzero homogeneous plurisubharmonic polynomial on \(\mathbb C^2\). We prove that every irreducible affine algebraic curve avoiding the origin on which \(p\) vanishes is exactly a nonzero level of a homogeneous holomorphic polynomial of degree at most \(\deg p\). No smoothness or rationality assumption is needed.
The proof uses a finite-pole rigidity lemma on the compact normalization: Levi positivity eliminates polar leading terms, and constancy along complex dilations yields a holomorphic level by polarization and binary-form irreducibility.
If reducible curves are allowed, a degree-eight counterexample exists even without pluriharmonic terms. The two curve conventions are explicitly distinguished.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Zero-curve theorem and counterexample
- Categories
- math.CV · math.AG
- Manuscript
- 5 October 2026
- Online release
- 5 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Algebraic Zero Curves of Homogeneous Plurisubharmonic Polynomials,” EulerSolve Research Papers, AIM-SEVERAL_COMPLEX_VARIABLES-0028, 2026. https://doi.org/10.5281/zenodo.23162700.
BibTeX
@misc{Ferudun2026AlgebraicZeroCurves,
author = {Ferudun, Alper},
title = {Algebraic Zero Curves of Homogeneous Plurisubharmonic Polynomials},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-several_complex_variables-0028/},
doi = {10.5281/zenodo.23162700},
note = {AIM-SEVERAL_COMPLEX_VARIABLES-0028; unrefereed preprint}
}