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Abstract

On the complex zero-sum hyperplane in ℂ^n, consider the symmetric quartics A=n∑_i|v_i|^4-|∑_i v_i^2|^2 and Q=‖v‖^4-|∑_i v_i^2|^2. For every odd n≥3 and every α,β>0, we prove that a holomorphic squared-norm representation of ‖v‖^(2m)(α A+β Q) 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.