Manuscript Em01-G@5 · Em01-G-v5.tex · proposition · 3fbfe951

Proposition: Highest normal polynomial

Supported algebraic coefficients · lines 629–664 · proof included or linked

Em01-G@5#prop:top, Em01-G@5#eq:top

The degree-three normal polynomial is

∣ ζ ∣ + j = 3 \sum c_{ζ, j}^{I, J} z^{ζ} w^{j} = m w [\frac{f ( b ) β}{4} ℓ_{μ} ℓ_{ν} + \frac{g ( b )}{2} (\partial_{μ} F) (b) ℓ_{ν}] Q^{I, J} (z, - z - h w)^{μν} .

In particular $c_{0, 3}^{I, J} = 0$ , and there are no pure residual degree-three jets.

Proof. The highest normal symbols of the response seed fields are

Z_{μ} \mapsto z_{μ}, A_{μ} \mapsto - z_{μ} - h_{μ} w, \partial_{s^{μ}} \mapsto h_{μ} w .

The order-three insertion sign is negative. The identity $A^{ρ μ}_{L} P^{L}_{M} = - (e_{M})^{ρ μ}$ cancels it. The $K_{8}$ multiplier is $s_{ρ} /2 = β ℓ_{ρ} /2$ . The other degree-three row is the derivative-root row. This gives (eq:top). Set $z = 0$ and use (eq:constant-slot); the factor $w$ excludes pure residual terms.

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Em01-G@5#eq:top · line 632
$∣ ζ ∣ + j = 3 \sum c_{ζ, j}^{I, J} z^{ζ} w^{j} = m w [\frac{f ( b ) β}{4} ℓ_{μ} ℓ_{ν} + \frac{g ( b )}{2} (\partial_{μ} F) (b) ℓ_{ν}] Q^{I, J} (z, - z - h w)^{μν} .$
unlabeled · line 650
$Z_{μ} \mapsto z_{μ}, A_{μ} \mapsto - z_{μ} - h_{μ} w, \partial_{s^{μ}} \mapsto h_{μ} w .$

Proposition: Highest normal polynomial

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