Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanes
- Primary subject
- Mathematics
- Subject subfields
- Complex Analysis · Algebra · Linear Algebra and Matrix Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
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.