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97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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