| import OAI.Combinatorics.CompleteCrossing.Isotropy |
| import OAI.Combinatorics.CompleteCrossing.TriangleFactors |
| import OAI.Combinatorics.CompleteCrossing.VertexEvaluation |
|
|
| namespace OAI |
|
|
| /-! Polynomial injectivity and the lower bound on crossing-tensor support. -/ |
|
|
| section |
|
|
| /-! The polynomial lower-bound argument for complete graphs. -/ |
| noncomputable section |
| namespace Paper170 |
| section CrossingTensor |
|
|
| variable {s : ℕ} (hs : 2 ≤ s) (node : Fin (2 * s + 1) → ℝ) |
| (hnode : Function.Injective node) (J : CrossingTensor (2 * s + 1)) |
| (orders : ∀ i : Fin (2 * s + 1), LinearOrder {j : Fin (2 * s + 1) // j ≠ i}) |
| (hloopLeft : ∀ i k q, J i i k q = 0) |
| (hloopRight : ∀ i p k, J i p k k = 0) |
| (hreverseLeft : ∀ i p k q, J p i k q = -J i p k q) |
| (hreverseRight : ∀ i p k q, J i p q k = -J i p k q) |
| (hskew : ∀ i p k q, J k q i p = -J i p k q) |
| (hlocal : ∀ i (p q : {j : Fin (2 * s + 1) // j ≠ i}), |
| J i p.val i q.val = halfOrder (orders i) p q) |
| (htests : ∀ H ∈ box4 (2 * s - 1), polynomialTest node J H = 0) |
|
|
| include hs hnode orders hloopLeft hloopRight hreverseLeft hreverseRight hskew hlocal htests |
|
|
| theorem kernel_diagonal_vanishing (P : endpointSymmetricSpace s) |
| (hP : supportEvaluation s node J P = 0) : |
| diagonalSpecialization P.val = 0 := |
| diagonal_zero_of_vertices s node hnode P (kernel_vertex_vanishing hs node hnode J orders |
| hloopLeft hloopRight hreverseLeft hreverseRight hskew hlocal htests P hP) |
|
|
| theorem kernel_collision_divisibility (P : endpointSymmetricSpace s) |
| (hP : supportEvaluation s node J P = 0) : collisionFactor ∣ P.val := |
| collision_divides_of_diagonal_vanishing s P (kernel_diagonal_vanishing hs node hnode J orders |
| hloopLeft hloopRight hreverseLeft hreverseRight hskew hlocal htests P hP) |
|
|
| theorem support_evaluation_injective : Function.Injective (supportEvaluation s node J) := |
| support_evaluation_injective_of_collision hs node hnode J |
| (kernel_collision_divisibility hs node hnode J orders hloopLeft hloopRight |
| hreverseLeft hreverseRight hskew hlocal htests) |
|
|
| theorem support_cardinality_lower_bound : |
| 2 * (Nat.choose s 2) ^ 2 ≤ (supportedPairs J).card := |
| support_cardinality_of_injective_isotropic hs node hnode J |
| (support_evaluation_injective hs node hnode J orders hloopLeft hloopRight |
| hreverseLeft hreverseRight hskew hlocal htests) |
| (evaluation_image_isotropic hs node hnode J orders hloopLeft hloopRight |
| hreverseLeft hreverseRight hskew hlocal htests) |
|
|
| end CrossingTensor |
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
