Manuscript XV-C@1 · XV-C.tex · proposition · 5afd124d

Proposition: Collected coefficients

The supplied supported coefficients through order three › Exact collection of the source formula · lines 736–798 · proof included or linked

XV-C@1#prop:collected, XV-C@1#eq:collected, XV-C@1#eq:degree3, XV-C@1#eq:degree2, XV-C@1#eq:degree1, XV-C@1#eq:degree0

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The unique coefficients in the supplied normal chart are

b_{D, γ}^{\circ} = β \geq 0 ∣ γ + β ∣ \leq 3 \sum (- 1)^{∣ β ∣} (γ γ + β) v_{γ + β} c_{D, β} .

In particular:

∣ γ ∣ = 3 : b_{D, γ}^{\circ} ∣ γ ∣ = 2 : b_{D, γ}^{\circ} ∣ γ ∣ = 1 : b_{D, γ}^{\circ} b_{D, 0}^{\circ} = v_{γ} c_{D, 0}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}} + ∣ β ∣ = 2 \sum (γ γ + β) v_{γ + β} c_{D, β}, = ∣ α ∣ \leq 3 \sum (- 1)^{∣ α ∣} v_{α} c_{D, α} .

Proof. In (eq:reference), put $γ = α - β$ . Then $α = γ + β$ and

(β α) = (γ γ + β) .

This gives (eq:collected). The displayed degree formulas are the finite sums according to $∣ β ∣ \leq 3 - ∣ γ ∣$ .

Uniqueness is Lemma lem:extraction, applied locally on the source domain.

Displayed formulas in this passage

XV-C@1#eq:collected · line 739
$b_{D, γ}^{\circ} = β \geq 0 ∣ γ + β ∣ \leq 3 \sum (- 1)^{∣ β ∣} (γ γ + β) v_{γ + β} c_{D, β} .$
XV-C@1#eq:degree3 · line 749
$∣ γ ∣ = 3 : b_{D, γ}^{\circ} ∣ γ ∣ = 2 : b_{D, γ}^{\circ} ∣ γ ∣ = 1 : b_{D, γ}^{\circ} b_{D, 0}^{\circ} = v_{γ} c_{D, 0}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}} + ∣ β ∣ = 2 \sum (γ γ + β) v_{γ + β} c_{D, β}, = ∣ α ∣ \leq 3 \sum (- 1)^{∣ α ∣} v_{α} c_{D, α} .$
XV-C@1#eq:degree2 · line 749
$∣ γ ∣ = 3 : b_{D, γ}^{\circ} ∣ γ ∣ = 2 : b_{D, γ}^{\circ} ∣ γ ∣ = 1 : b_{D, γ}^{\circ} b_{D, 0}^{\circ} = v_{γ} c_{D, 0}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}} + ∣ β ∣ = 2 \sum (γ γ + β) v_{γ + β} c_{D, β}, = ∣ α ∣ \leq 3 \sum (- 1)^{∣ α ∣} v_{α} c_{D, α} .$
XV-C@1#eq:degree1 · line 749
$∣ γ ∣ = 3 : b_{D, γ}^{\circ} ∣ γ ∣ = 2 : b_{D, γ}^{\circ} ∣ γ ∣ = 1 : b_{D, γ}^{\circ} b_{D, 0}^{\circ} = v_{γ} c_{D, 0}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}} + ∣ β ∣ = 2 \sum (γ γ + β) v_{γ + β} c_{D, β}, = ∣ α ∣ \leq 3 \sum (- 1)^{∣ α ∣} v_{α} c_{D, α} .$
XV-C@1#eq:degree0 · line 749
$∣ γ ∣ = 3 : b_{D, γ}^{\circ} ∣ γ ∣ = 2 : b_{D, γ}^{\circ} ∣ γ ∣ = 1 : b_{D, γ}^{\circ} b_{D, 0}^{\circ} = v_{γ} c_{D, 0}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}}, = v_{γ} c_{D, 0} - j = 1 \sum^{m} (γ_{j} + 1) v_{γ + e_{j}} c_{D, e_{j}} + ∣ β ∣ = 2 \sum (γ γ + β) v_{γ + β} c_{D, β}, = ∣ α ∣ \leq 3 \sum (- 1)^{∣ α ∣} v_{α} c_{D, α} .$
unlabeled · line 787
$(β α) = (γ γ + β) .$

Proposition: Collected coefficients

Displayed formulas in this passage

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