File size: 5,443 Bytes
7d9acbd | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import OAI.Geometry.HeilbronnTriangle.ZModAnnihilator
namespace OAI
namespace Problem355.DiagonalKernel
open scoped BigOperators Matrix
variable {R ι : Type*} [Semiring R] [Fintype ι] [DecidableEq ι]
abbrev Kernel (A : Matrix ι ι R) := {x : ι → R // A.mulVec x = 0}
def diagonalKernelEquiv (d : ι → R) :
Kernel (Matrix.diagonal d) ≃ (∀ i, {x : R // d i * x = 0}) where
toFun x i := ⟨x.1 i, by
have h := congrFun x.2 i
simpa only [Matrix.mulVec_diagonal, Pi.zero_apply] using h⟩
invFun x := ⟨fun i => (x i).1, by
funext i
simpa only [Matrix.mulVec_diagonal, Pi.zero_apply] using (x i).2⟩
left_inv x := by rfl
right_inv x := by rfl
def unitMulVecEquiv (Q : (Matrix ι ι R)ˣ) : (ι → R) ≃ (ι → R) where
toFun := (Q : Matrix ι ι R).mulVec
invFun := ((Q⁻¹ : (Matrix ι ι R)ˣ) : Matrix ι ι R).mulVec
left_inv x := by
rw [Matrix.mulVec_mulVec]
simp
right_inv x := by
rw [Matrix.mulVec_mulVec]
simp
def changeKernelEquiv (A : Matrix ι ι R) (P Q : (Matrix ι ι R)ˣ) :
Kernel ((P : Matrix ι ι R) * A * (Q : Matrix ι ι R)) ≃ Kernel A := by
apply Equiv.subtypeEquiv (unitMulVecEquiv Q)
intro x
change (((P : Matrix ι ι R) * A * (Q : Matrix ι ι R)).mulVec x = 0) ↔
A.mulVec ((Q : Matrix ι ι R).mulVec x) = 0
rw [← Matrix.mulVec_mulVec, ← Matrix.mulVec_mulVec]
exact (Matrix.mulVec_injective_of_isUnit P.isUnit).eq_iff' (by simp)
theorem natCard_kernel_change (A : Matrix ι ι R) (P Q : (Matrix ι ι R)ˣ) :
Nat.card (Kernel ((P : Matrix ι ι R) * A * (Q : Matrix ι ι R))) =
Nat.card (Kernel A) :=
Nat.card_congr (changeKernelEquiv A P Q)
section Cardinality
variable [Fintype R] [DecidableEq R]
theorem card_kernel_diagonal (d : ι → R) :
Fintype.card (Kernel (Matrix.diagonal d)) =
∏ i, Fintype.card {x : R // d i * x = 0} := by
classical
rw [Fintype.card_congr (diagonalKernelEquiv d), Fintype.card_pi]
theorem card_kernel_change (A : Matrix ι ι R) (P Q : (Matrix ι ι R)ˣ) :
Fintype.card (Kernel ((P : Matrix ι ι R) * A * (Q : Matrix ι ι R))) =
Fintype.card (Kernel A) :=
Fintype.card_congr (changeKernelEquiv A P Q)
theorem card_kernel_diagonal_three (D E : R) :
Fintype.card (Kernel (Matrix.diagonal ![1, D, E])) =
Fintype.card {x : R // D * x = 0} *
Fintype.card {x : R // E * x = 0} := by
rw [card_kernel_diagonal]
simp [Fin.prod_univ_succ]
exact Or.inl rfl
end Cardinality
theorem card_pow_annihilator (B j b : ℕ) [NeZero B] :
Nat.card {x : ZMod (B ^ j) // (B : ZMod (B ^ j)) ^ b * x = 0} =
B ^ min b j := by
rcases le_total b j with h | h
· simpa only [Nat.min_eq_left h] using zmod_card_pow_mul_eq_zero B j b h
· simpa only [Nat.cast_pow, Nat.gcd_eq_left (pow_dvd_pow B h),
Nat.min_eq_right h] using zmod_card_mul_eq_zero (B ^ j) (B ^ b)
theorem card_kernel_diagonal_prime_power (B j b e : ℕ) [NeZero B]
(hje : j ≤ e) :
Nat.card (Kernel (Matrix.diagonal
![1, (B : ZMod (B ^ j)) ^ b, (B : ZMod (B ^ j)) ^ e])) =
B ^ (j + min b j) := by
classical
rw [Nat.card_eq_fintype_card, card_kernel_diagonal_three,
← Nat.card_eq_fintype_card, ← Nat.card_eq_fintype_card,
card_pow_annihilator, card_pow_annihilator,
Nat.min_eq_right hje, pow_add]
exact Nat.mul_comm _ _
theorem card_kernel_diagonal_prime_power_density (B j b e : ℕ) [NeZero B]
(hje : j ≤ e) :
Nat.card (Kernel (Matrix.diagonal
![1, (B : ZMod (B ^ j)) ^ b, (B : ZMod (B ^ j)) ^ e])) *
B ^ (j + (j - b)) = (B ^ j) ^ 3 := by
rw [card_kernel_diagonal_prime_power B j b e hje, ← pow_add, ← pow_mul]
congr 1
omega
theorem card_kernel_prime_power_density (B j b e : ℕ) [NeZero B]
(hje : j ≤ e) (C : Matrix (Fin 3) (Fin 3) (ZMod (B ^ j)))
(P Q : (Matrix (Fin 3) (Fin 3) (ZMod (B ^ j)))ˣ)
(hC : C = (P : Matrix _ _ _) *
Matrix.diagonal ![1, (B : ZMod (B ^ j)) ^ b, (B : ZMod (B ^ j)) ^ e] *
(Q : Matrix _ _ _)) :
Nat.card (Kernel C) * B ^ (j + (j - b)) = (B ^ j) ^ 3 := by
rw [hC, natCard_kernel_change]
exact card_kernel_diagonal_prime_power_density B j b e hje
theorem card_kernel_reduction_density (B k j b e : ℕ) [NeZero B]
(hjk : j ≤ k) (hje : j ≤ e)
(C : Matrix (Fin 3) (Fin 3) (ZMod (B ^ k)))
(P Q : (Matrix (Fin 3) (Fin 3) (ZMod (B ^ k)))ˣ)
(hC : C = (P : Matrix _ _ _) *
Matrix.diagonal ![1, (B : ZMod (B ^ k)) ^ b, (B : ZMod (B ^ k)) ^ e] *
(Q : Matrix _ _ _)) :
Nat.card (Kernel (C.map
(ZMod.castHom (pow_dvd_pow B hjk) (ZMod (B ^ j))))) *
B ^ (j + (j - b)) = (B ^ j) ^ 3 := by
let f : ZMod (B ^ k) →+* ZMod (B ^ j) :=
ZMod.castHom (pow_dvd_pow B hjk) (ZMod (B ^ j))
have hd : (Matrix.diagonal
![1, (B : ZMod (B ^ k)) ^ b, (B : ZMod (B ^ k)) ^ e]).map f =
Matrix.diagonal ![1, (B : ZMod (B ^ j)) ^ b, (B : ZMod (B ^ j)) ^ e] := by
ext i l
by_cases h : i = l
· subst l
fin_cases i <;> simp
· simp [h]
apply card_kernel_prime_power_density B j b e hje (C.map f)
(Units.map f.mapMatrix.toMonoidHom P) (Units.map f.mapMatrix.toMonoidHom Q)
change C.map f = (P : Matrix _ _ _).map f *
Matrix.diagonal ![1, (B : ZMod (B ^ j)) ^ b, (B : ZMod (B ^ j)) ^ e] *
(Q : Matrix _ _ _).map f
simpa only [Matrix.map_mul, hd] using congrArg (fun A => A.map f) hC
end Problem355.DiagonalKernel
end OAI
|