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Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Complex Analysis
  • Algebra
  • Linear Algebra and Matrix Theory
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

On the complex zero-sum hyperplane in C^n, consider the symmetric quartics A = n∑ᵢ|vᵢ|⁴ − |∑ᵢvᵢ²|² and Q = ‖v‖⁴ − |∑ᵢvᵢ²|². For every odd n ≥ 3 and every α, β > 0, we prove that a holomorphic squared-norm representation of ‖v‖²ᵐ(αA + βQ) requires m ≥ n − 2. For each fixed odd n, equality holds for all sufficiently small positive β/α, whereas at β = 0 the least exponent is zero. The lower bound uses an explicit Fischer annihilator and a positive Hamming-distance correlation identity on the odd-dimensional sign cube. The upper bound uses the classical quadratic complete intersection associated with the zero-sum hyperplane. For n = 5, we determine the entire first-degree window: the least exponent is three exactly when 0 < β/α ≤ ρ*, where 305ρ*³ + 786ρ*² + 351ρ* − 162 = 0. An explicit four-by-four pencil and harmonic decomposition give this sharp endpoint, including its four-dimensional kernel. We also derive the quartic as a coefficient in a centered permanent deficit, which motivates the family without implying the full permanent inequality.

Keywords Hermitian polynomial positivity · holomorphic squared norms · Fischer product · stabilization degree · zero-sum hyperplanes

Scope of the work

Fixed quartic family on odd-coordinate complex zero-sum hyperplanes: uniform degree lower bound m ≥ n − 2; finite first-degree windows characterized spectrally; explicit sharp five-coordinate endpoint. The general threshold is implicit. No minimum-degree classification above the first window, worldwide-priority assertion, or solution of the generic rank-six Chollet compensation problem is claimed.

Reproduction notes

The supplementary eight-member package contains complete LaTeX source, reader instructions, citation metadata, a SHA-256 manifest and a standard-library Python verifier. Run python Reproducibility/verify_certificates.py and python -O Reproducibility/verify_certificates.py. The manuscript gives analytic all-odd proofs and an explicit four-by-four rational pencil; finite exact calculations are corroborative.

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Author disclosures

Research preprint. AI-assisted tools were used in mathematical exploration and manuscript preparation. Classical stabilization, the three-coordinate example, prior hierarchy results and the complete-intersection ideal are explicitly credited. Archive publication is not external peer review or journal acceptance.

References and related publications

  1. Catlin and D'Angelo (1996), A stabilization theorem for Hermitian forms
  2. Chmutova and Etingof (2003), Rational Cherednik algebra representations
  3. Halfpap and Lebl (2013), Signature pairs of positive polynomials
  4. Drouot and Zworski (2013), Quantitative stabilization
  5. Debus, Goel, Kuhlmann and Riener (2025), Type-A invariant quartics
  6. Schubr (2026), Occupation bounds for coupled permanent cofactors, edition 10

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/dimension-dependent-holomorphic-certificate-degrees-zero-sum-hyperplanes/v/1

BibTeX
@techreport{luminary_dimension_dependent_holomorphic_certificate_degrees_zero_sum_hyperplanes_v1,
  title = {Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/dimension-dependent-holomorphic-certificate-degrees-zero-sum-hyperplanes/v/1}
}

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