Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes
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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.
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- Certificate_degrees_zero_sum_hyperplanes.pdf116.3 KiB · pdf
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Research details
- Subjects
- Mathematics · Algebra · Linear Algebra and Matrix Theory · Complex Analysis
- Keywords
- Hermitian polynomial positivity, holomorphic squared norms, Fischer product, stabilization degree, zero-sum hyperplanes
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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.
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.
Reproduction
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.
References and related resources
- Catlin and D'Angelo (1996), A stabilization theorem for Hermitian forms
- Chmutova and Etingof (2003), Rational Cherednik algebra representations
- Halfpap and Lebl (2013), Signature pairs of positive polynomials
- Drouot and Zworski (2013), Quantitative stabilization
- Debus, Goel, Kuhlmann and Riener (2025), Type-A invariant quartics
- Schubr (2026), Occupation bounds for coupled permanent cofactors, edition 10