| import OAI.Combinatorics.CompleteCrossing.Polynomials |
| import Mathlib.LinearAlgebra.BilinearForm.Orthogonal |
|
|
| namespace OAI |
|
|
| /-! The support form, injectivity criterion, and support-space dimension. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
| open MvPolynomial |
| namespace Paper170 |
|
|
| theorem collision_nonzero_on_support {n : ℕ} (node : Fin n → ℝ) |
| (hn : Function.Injective node) (J : CrossingTensor n) (a : SupportCoord J) : |
| evalFour (node a.val.1.1) (node a.val.1.2) (node a.val.2.1) (node a.val.2.2) |
| collisionFactor ≠ 0 := by |
| rcases (Finset.mem_filter.mp a.property).2 with ⟨_, _, h02, h03, h12, h13, _⟩ |
| simpa [evalFour, collisionFactor, Matrix.cons_val_two, Matrix.cons_val_three] using |
| mul_ne_zero (mul_ne_zero (mul_ne_zero (sub_ne_zero.mpr (hn.ne h02)) |
| (sub_ne_zero.mpr (hn.ne h03))) (sub_ne_zero.mpr (hn.ne h12))) |
| (sub_ne_zero.mpr (hn.ne h13)) |
|
|
| lemma support_diagonal_nonzero {n : ℕ} (node : Fin n → ℝ) |
| (hn : Function.Injective node) (J : CrossingTensor n) (a : SupportCoord J) : |
| supportDiagonal node J a ≠ 0 := by |
| rcases (Finset.mem_filter.mp a.property).2 with ⟨h01, h23, _, _, _, _, hJ⟩ |
| have hc := collision_nonzero_on_support node hn J a |
| have hf : evalFour (node a.val.1.1) (node a.val.1.2) (node a.val.2.1) |
| (node a.val.2.2) fullDifference ≠ 0 := by |
| simpa [fullDifference, evalFour, Matrix.cons_val_two, Matrix.cons_val_three] using |
| mul_ne_zero (mul_ne_zero (sub_ne_zero.mpr (hn.ne (ne_of_lt h01))) |
| (sub_ne_zero.mpr (hn.ne (ne_of_lt h23)))) hc |
| exact mul_ne_zero (mul_ne_zero (mul_ne_zero (mul_ne_zero (mul_ne_zero hJ |
| (weight_nonzero node hn _)) (weight_nonzero node hn _)) |
| (weight_nonzero node hn _)) (weight_nonzero node hn _)) hf |
|
|
| theorem support_bilinear_nondegenerate {n : ℕ} (node : Fin n → ℝ) |
| (hn : Function.Injective node) (J : CrossingTensor n) : |
| (supportBilinear node J).Nondegenerate := by |
| apply LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero.mpr |
| rw [Matrix.det_diagonal] |
| exact Finset.prod_ne_zero_iff.mpr fun a _ => support_diagonal_nonzero node hn J a |
|
|
| theorem isotropic_dimension {ι : Type*} [Fintype ι] |
| (B : LinearMap.BilinForm ℝ (ι → ℝ)) (hB : B.Nondegenerate) |
| (W : Submodule ℝ (ι → ℝ)) (hW : W ≤ B.orthogonal W) : |
| 2 * Module.finrank ℝ W ≤ Fintype.card ι := by |
| have hle := Submodule.finrank_mono hW |
| rw [LinearMap.BilinForm.finrank_orthogonal hB] at hle |
| have hdim := Submodule.finrank_le W |
| have hv : Module.finrank ℝ (ι → ℝ) = Fintype.card ι := by simp |
| rw [hv] at hle hdim |
| omega |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| @[simp] lemma evalFour_mul (x u y v : ℝ) (P Q : Poly4) : |
| evalFour x u y v (P * Q) = evalFour x u y v P * evalFour x u y v Q := |
| map_mul (MvPolynomial.aeval ![x, u, y, v]) P Q |
|
|
| lemma support_evaluation_injective_of_collision {n s : ℕ} (hs : 2 ≤ s) |
| (node : Fin n → ℝ) (hn : Function.Injective node) (J : CrossingTensor n) |
| (hdiv : ∀ P : endpointSymmetricSpace s, |
| supportEvaluation s node J P = 0 → collisionFactor ∣ P.val) : |
| Function.Injective (supportEvaluation s node J) := by |
| have hzero : ∀ d (P : endpointSymmetricSpace s), P.val.degreeOf 0 = d → |
| supportEvaluation s node J P = 0 → P.val = 0 := by |
| intro d |
| induction d using Nat.strong_induction_on with |
| | h d ih => |
| intro P hd hP |
| by_contra hP0 |
| obtain ⟨Q, hQ0, heq, hQmem, _, hQdeg⟩ := collision_quotient s hs P.val |
| P.property hP0 (hdiv P hP) |
| have hQ : supportEvaluation s node J ⟨Q, hQmem⟩ = 0 := by |
| funext a |
| have hp := congrFun hP a |
| change evalFour (node a.val.1.1) (node a.val.1.2) |
| (node a.val.2.1) (node a.val.2.2) P.val = 0 at hp |
| rw [heq, evalFour_mul] at hp |
| exact (mul_eq_zero.mp hp).resolve_left (collision_nonzero_on_support node hn J a) |
| exact hQ0 (ih (Q.degreeOf 0) (by omega) ⟨Q, hQmem⟩ rfl hQ) |
| rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot'] |
| intro P hP |
| exact Subtype.ext (hzero _ P rfl hP) |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| lemma support_cardinality_of_injective_isotropic {n s : ℕ} (hs : 2 ≤ s) |
| (node : Fin n → ℝ) (hn : Function.Injective node) (J : CrossingTensor n) |
| (hi : Function.Injective (supportEvaluation s node J)) |
| (ho : ∀ P Q : endpointSymmetricSpace s, |
| supportBilinear node J (supportEvaluation s node J P) (supportEvaluation s node J Q) = 0) : |
| 2 * (Nat.choose s 2) ^ 2 ≤ (supportedPairs J).card := by |
| let := symmetric_space_finite s |
| let f := supportEvaluation s node J |
| have hr : f.range ≤ (supportBilinear node J).orthogonal f.range := by |
| rintro _ ⟨P, rfl⟩ _ ⟨Q, rfl⟩ |
| exact ho Q P |
| have hd := isotropic_dimension (supportBilinear node J) |
| (support_bilinear_nondegenerate node hn J) f.range hr |
| have he : Module.finrank ℝ f.range = Module.finrank ℝ (endpointSymmetricSpace s) := |
| (LinearEquiv.ofInjective f hi).finrank_eq.symm |
| rw [he, symmetric_space_dimension s hs, Fintype.card_coe] at hd |
| exact hd |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
