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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 | import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Separation.Basic
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Character.Degeneration
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Character.Dot
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Tensor.CharacterBounds
namespace OAI
/-!
# The character bound from finite separation
The explicit Fourier and square-weight construction gives a polynomial tensor
whose nonzero evaluations restrict from exactly `5M` complete source copies.
Character monotonicity follows by interpolation and tensor powering; it does
not require continuity of the character.
-/
noncomputable section
open MatrixMultiplication.Foundation
open scoped BigOperators
namespace MatrixMultiplication.AuxiliarySeparation
variable {M : ℕ} {X Y Z : Type}
variable [Fintype X] [Fintype Y] [Fintype Z]
omit [Fintype Z] in
private theorem polynomial_rank_bound
(P : Tensor (Polynomial ℂ) X Y Z) :
Tensor.RankAtMost P (Fintype.card (X × Y)) := by
classical
let a : X × Y → X → Polynomial ℂ := fun i x => if i.1 = x then 1 else 0
let b : X × Y → Y → Polynomial ℂ := fun i y => if i.2 = y then 1 else 0
let c : X × Y → Z → Polynomial ℂ := fun i z => P i.1 i.2 z
have heq : P = fun x y z => ∑ i, Tensor.rankOne (a i) (b i) (c i) x y z := by
funext x y z
simp [Tensor.rankOne, a, b, c, Fintype.sum_prod_type, ite_mul, mul_ite]
rw [heq]
exact Tensor.rankAtMost_sum_rankOne a b c
/-- The explicit separation polynomial, viewed as a polynomial approximation
to its constant tensor. Its rank bound is supplied by a coordinate expansion. -/
def separationPolynomialApproximation {ζ : ℂ} (hM : 0 < M)
(hζ : IsPrimitiveRoot ζ (5 * M)) (B : Fin M → Tensor ℂ X Y Z) :
Tensor.PolynomialApproximation (separationTarget B)
(Fintype.card ((X × Fin M) × ((Fin M × Y) × Fin M))) 0 ((M - 1) ^ 2) where
polynomial := separationPolynomial ζ B
rank_bound := polynomial_rank_bound _
vanishes := by
intro _ _ _ k hk
exact (Nat.not_lt_zero k hk).elim
leading := by
intro x y z
exact congrFun (congrFun (congrFun (separationPolynomial_coeff_zero hM hζ B) x) y) z
degree_bound := separationPolynomial_degree hM hζ B
namespace Character
variable (χ : Character)
/-- The separated constant tensor costs at most the character of the `5M`
source copies. This uses the explicit polynomial, not topological continuity. -/
theorem value_separationTarget_le (hM : 0 < M) (B : Fin M → Tensor ℂ X Y Z) :
χ.value (separationTarget B) ≤
5 * (M : ℝ) * χ.value (sharedFirstTensor B) := by
classical
let ζ : ℂ := Complex.exp (2 * Real.pi * Complex.I / (5 * M : ℕ))
have hζ : IsPrimitiveRoot ζ (5 * M) :=
Complex.isPrimitiveRoot_exp _ (by omega)
have hbound : χ.value (separationTarget B) ≤
χ.value (Tensor.directSum (fun _ : Fin (5 * M) => sharedFirstTensor B)) := by
apply χ.value_le_of_polynomialApproximation
(separationPolynomialApproximation hM hζ B) (χ.nonneg _)
intro t ht
change χ.value (fun x y z => (separationPolynomial ζ B x y z).eval t) ≤ _
rw [separationPolynomial_eval hM hζ B t ht]
exact χ.monotone _ _ _ _
simpa only [χ.value_copies, Nat.cast_mul, Nat.cast_ofNat] using hbound
/-- Character accounting for the genuine direct sum in the separated tensor. -/
theorem value_separationTarget (hM : 0 < M) (B : Fin M → Tensor ℂ X Y Z) :
χ.value (separationTarget B) =
(∑ h, χ.value (B h)) * (M : ℝ) ^ χ.pX := by
classical
rw [separationTarget_eq_directSum, χ.value_reindex, χ.map_directSum]
simp only [χ.map_product, χ.value_cyclic_cyclic_dotPairing hM, Finset.sum_mul]
/-- Exact finite character separation: the common first coordinate can be
separated at a cost of `5M`, retaining an `M`-dimensional first-leg dot factor. -/
theorem finiteSeparation_bound (hM : 0 < M) (B : Fin M → Tensor ℂ X Y Z) :
(∑ h, χ.value (B h)) * (M : ℝ) ^ χ.pX ≤
5 * (M : ℝ) * χ.value (sharedFirstTensor B) := by
rw [← χ.value_separationTarget hM B]
exact χ.value_separationTarget_le hM B
end Character
end MatrixMultiplication.AuxiliarySeparation
end
end OAI
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