openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Determinant /Basis.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Determinant.Kernel | |
| import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | |
| namespace OAI | |
| /-! | |
| # The finite adapted basis for the determinant filtration | |
| The quotient vectors and determinant-kernel vectors of Section 5.1 form a basis | |
| of the actual space of bidegree `(e,1)` forms. The proof works over every field, | |
| including in the boundary case `e = 0` where the kernel has dimension zero. | |
| -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AuxiliarySeparation.DeterminantBasis | |
| open Polynomial | |
| variable {R : Type*} [CommRing R] | |
| /-- Affine coefficient pairs of forms of bidegree `(e,1)`. -/ | |
| abbrev BoundedPair (R : Type*) [CommRing R] (e : ℕ) := | |
| degreeLT R (e + 1) × degreeLT R (e + 1) | |
| /-- Forget the degree bounds in the affine representation of binary forms. -/ | |
| def forget (e : ℕ) : BoundedPair R e →ₗ[R] DeterminantKernel.FormPair R where | |
| toFun p := ((p.1 : R[X]), (p.2 : R[X])) | |
| map_add' _ _ := rfl | |
| map_smul' _ _ := rfl | |
| theorem forget_injective (e : ℕ) : Function.Injective (forget (R := R) e) := by | |
| intro p q h | |
| apply Prod.ext <;> apply Subtype.ext | |
| · exact congrArg Prod.fst h | |
| · exact congrArg Prod.snd h | |
| /-- A quotient vector with both coefficient polynomials of degree at most `e`. -/ | |
| def quotientVector (e : ℕ) (j : Fin (e + 2)) : BoundedPair R e := | |
| if h : (j : ℕ) < e + 1 then | |
| (⟨X ^ (j : ℕ), mem_degreeLT.mpr ((degree_X_pow_le _).trans_lt (by exact_mod_cast h))⟩, 0) | |
| else | |
| (0, ⟨X ^ e, mem_degreeLT.mpr ((degree_X_pow_le _).trans_lt (by exact_mod_cast Nat.lt_succ_self e))⟩) | |
| /-- A bounded kernel vector, obtained by multiplication by `uw-vs`. -/ | |
| def kernelVector (e : ℕ) (j : Fin e) : BoundedPair R e := | |
| (⟨-(X ^ ((j : ℕ) + 1)), mem_degreeLT.mpr <| by | |
| rw [degree_neg] | |
| exact (degree_X_pow_le _).trans_lt (by exact_mod_cast (show (j : ℕ) + 1 < e + 1 by omega))⟩, | |
| ⟨X ^ (j : ℕ), mem_degreeLT.mpr <| | |
| (degree_X_pow_le _).trans_lt (by exact_mod_cast (show (j : ℕ) < e + 1 by omega))⟩) | |
| /-- The quotient vectors first, followed by the determinant-kernel vectors. -/ | |
| def vector (e : ℕ) : (Fin (e + 2) ⊕ Fin e) → BoundedPair R e := | |
| Sum.elim (quotientVector e) (kernelVector e) | |
| @[simp] theorem forget_quotientVector (e : ℕ) (j : Fin (e + 2)) : | |
| forget e (quotientVector (R := R) e j) = DeterminantKernel.quotientVector e j := by | |
| unfold quotientVector DeterminantKernel.quotientVector | |
| split_ifs <;> rfl | |
| @[simp] theorem forget_kernelVector (e : ℕ) (j : Fin e) : | |
| forget e (kernelVector (R := R) e j) = DeterminantKernel.kernelVector e j := by | |
| apply Prod.ext | |
| · change -(X ^ ((j : ℕ) + 1)) = -X * X ^ (j : ℕ) | |
| rw [pow_succ, mul_comm, neg_mul] | |
| · rfl | |
| @[simp] theorem forget_vector (e : ℕ) (j : Fin (e + 2) ⊕ Fin e) : | |
| forget e (vector (R := R) e j) = | |
| Sum.elim (DeterminantKernel.quotientVector e) (DeterminantKernel.kernelVector e) j := by | |
| cases j <;> simp [vector] | |
| /-- Linear independence holds over a general commutative coefficient ring. -/ | |
| theorem vector_linearIndependent (e : ℕ) : LinearIndependent R (vector (R := R) e) := by | |
| apply LinearIndependent.of_comp (forget e) | |
| simpa only [Function.comp_def, forget_vector] using | |
| DeterminantKernel.adaptedVectors_linearIndependent (R := R) e | |
| /-- The dimension of the original space of forms is twice `e+1`. -/ | |
| theorem finrank_boundedPair (K : Type*) [Field K] (e : ℕ) : | |
| Module.finrank K (BoundedPair K e) = 2 * (e + 1) := by | |
| have h : Module.finrank K (degreeLT K (e + 1)) = e + 1 := by | |
| simpa using Module.finrank_eq_card_basis (degreeLT.basis K (e + 1)) | |
| rw [Module.finrank_prod, h] | |
| omega | |
| /-- The fixed simultaneous change of basis used in the determinant degeneration. -/ | |
| def basis (K : Type*) [Field K] (e : ℕ) : | |
| Module.Basis (Fin (e + 2) ⊕ Fin e) K (BoundedPair K e) := | |
| basisOfLinearIndependentOfCardEqFinrank' (vector e) (vector_linearIndependent e) (by | |
| rw [Fintype.card_sum, Fintype.card_fin, Fintype.card_fin, finrank_boundedPair] | |
| omega) | |
| @[simp] theorem basis_apply (K : Type*) [Field K] (e : ℕ) | |
| (j : Fin (e + 2) ⊕ Fin e) : basis K e j = vector e j := by | |
| rw [basis, coe_basisOfLinearIndependentOfCardEqFinrank'] | |
| /-- Forgetting degree bounds identifies the finite basis with the explicit | |
| polynomial-pair formulas. -/ | |
| @[simp] theorem forget_basis (K : Type*) [Field K] (e : ℕ) | |
| (j : Fin (e + 2) ⊕ Fin e) : | |
| forget e (basis K e j) = | |
| Sum.elim (DeterminantKernel.quotientVector e) (DeterminantKernel.kernelVector e) j := by | |
| rw [basis_apply, forget_vector] | |
| end MatrixMultiplication.AuxiliarySeparation.DeterminantBasis | |
| end | |
| end OAI | |