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cdec422 | 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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | 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
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