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a72760f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupGeometry
import OAI.NumberTheory.PiExponent.Approximation.DeterminantContradiction
import OAI.NumberTheory.PiExponent.Approximation.FormalMatrixSurjectivity
import OAI.NumberTheory.PiExponent.Approximation.WeightedSliceDegree
namespace OAI
noncomputable section
namespace PiExponent.AdmissibleMatrixInterpolation
open Filter
open DeterminantContradiction AdmissibleBlowupGeometry
variable {ν : ℝ} (d : FixedData ν)
theorem degreeWeights_eq_matrix (i : Fin (d.m+1)) :
(d.curveDegreeWeights i : ℝ) =
InterpolationMatrix.columnWeights d.w0
(MatrixArithmetic.logWeights (finiteDenominators d)) i := by
refine Fin.cases ?_ (fun j => ?_) i
· rfl
· simp only [AdmissibleParameters.curveDegreeWeights_succ,
d.cast_rationalWeight, InterpolationMatrix.columnWeights, Fin.cases_succ,
logWeights_eq]
theorem jetWeights_eq_matrix (i : Fin (d.m+1)) :
(d.curveJetWeights i : ℝ) =
InterpolationMatrix.rowWeights d.v0 d.base.theta
(MatrixArithmetic.logWeights (finiteDenominators d)) i := by
refine Fin.cases ?_ (fun j => ?_) i
· rfl
· simp only [AdmissibleParameters.curveJetWeights_succ, Rat.cast_div,
d.cast_rationalWeight, InterpolationMatrix.rowWeights, Fin.cases_succ,
logWeights_eq]
theorem actual_truncation_weight_bound (i : Fin d.m) :
d.curveJetWeights i.succ ≤
(MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0 i : ℚ) *
d.curveJetWeights 0 := by
have hw : 0 < MatrixArithmetic.logWeights (finiteDenominators d) i := by
rw [logWeights_eq]
exact zero_lt_one.trans_le (d.x_one_le _)
have hF : 1 / (d.base.theta : ℝ) < d.F0 :=
(div_lt_div_of_pos_right (by norm_num : (1 : ℝ) < 2) d.base.theta_pos).trans d.F0_large
have hceil := Nat.le_ceil
(d.F0 * MatrixArithmetic.logWeights (finiteDenominators d) i / (d.v0 : ℝ))
have hbound : MatrixArithmetic.logWeights (finiteDenominators d) i / (d.base.theta : ℝ) ≤
(MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0 i : ℝ) *
(d.v0 : ℝ) := by
calc
_ = (1 / (d.base.theta : ℝ)) *
MatrixArithmetic.logWeights (finiteDenominators d) i := by ring
_ ≤ d.F0 * MatrixArithmetic.logWeights (finiteDenominators d) i :=
mul_le_mul_of_nonneg_right hF.le hw.le
_ ≤ _ := (div_le_iff₀ d.v0_pos).mp hceil
have hcast : (d.curveJetWeights i.succ : ℝ) ≤
(MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0 i : ℝ) *
(d.curveJetWeights 0 : ℝ) := by
simpa only [AdmissibleParameters.curveJetWeights_zero,
AdmissibleParameters.curveJetWeights_succ, Rat.cast_div,
d.cast_rationalWeight, logWeights_eq] using hbound
exact_mod_cast hcast
theorem matrix_degree_of_supportBound (H : ℝ) (P : PiExponentApprox.FramePolynomial d.m)
(hP : WeightedSliceDegree.SupportBound (fun i => (d.curveDegreeWeights i : ℝ)) H P) :
PiExponentApprox.HasWeightedDegreeLE
(InterpolationMatrix.columnWeights d.w0
(MatrixArithmetic.logWeights (finiteDenominators d))) H P := by
intro a ha
have h := hP a ha
have hw : (fun i => (d.curveDegreeWeights i : ℝ)) =
InterpolationMatrix.columnWeights d.w0
(MatrixArithmetic.logWeights (finiteDenominators d)) :=
funext (degreeWeights_eq_matrix d)
rw [hw] at h
simpa only [PiExponentApprox.monomialWeight, Finsupp.weight_eq_sum, nsmul_eq_mul] using h
theorem actualMatrix_surjective_of_formalPackets {α : Type*} (H : ℚ)
(P : α → PiExponentApprox.FramePolynomial d.m)
(hdegree : ∀ a, WeightedSliceDegree.SupportBound
(fun i => (d.curveDegreeWeights i : ℝ)) (H : ℝ) (P a))
(hpacket : Function.Surjective (fun a => fun j : Fin d.K =>
JetGeometry.rationalCoefficientPacket d.curveJetWeights H
(FormalLogJet.formalJet (d.curveCenters j) (P a)))) :
Function.Surjective (actualMatrix d (H : ℝ)).mulVecLin := by
apply FormalMatrixBridge.truncatedLogMatrix_surjective_of_formalLog_packets
d.K d.w0 d.v0 d.base.theta
(MatrixArithmetic.logWeights (finiteDenominators d)) H
d.w0_pos (fun i => ?_) d.curveJetWeights d.curveJetWeights_pos
(jetWeights_eq_matrix d)
(MatrixArithmetic.rationalCenters (finiteNumerators d) (finiteDenominators d))
(MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0)
(actual_truncation_weight_bound d) P
(fun a => matrix_degree_of_supportBound d _ _ (hdegree a)) hpacket
rw [logWeights_eq]
exact zero_lt_one.trans_le (d.x_one_le _)
def WeightedPolynomials (n : ℕ) :=
{P : PiExponentApprox.FramePolynomial d.m //
WeightedSliceDegree.SupportBound (fun i => (d.curveDegreeWeights i : ℝ))
((n : ℝ) * (scale d).radius) P}
def packetMap (n : ℕ) (P : WeightedPolynomials d n) :=
fun j : Fin d.K => JetGeometry.rationalCoefficientPacket d.curveJetWeights
((n : ℚ) * (scale d).radius) (FormalLogJet.formalJet (d.curveCenters j) P.val)
theorem packetMap_surjective_of_family {α : Type*} (n : ℕ)
(P : α → PiExponentApprox.FramePolynomial d.m)
(hdegree : ∀ a, WeightedSliceDegree.SupportBound
(fun i => (d.curveDegreeWeights i : ℝ)) ((n : ℝ) * (scale d).radius) (P a))
(hpacket : Function.Surjective (fun a => fun j : Fin d.K =>
JetGeometry.rationalCoefficientPacket d.curveJetWeights
((n : ℚ) * (scale d).radius) (FormalLogJet.formalJet (d.curveCenters j) (P a)))) :
Function.Surjective (packetMap d n) := by
intro y
obtain ⟨a,ha⟩ := hpacket y
exact ⟨⟨P a,hdegree a⟩,ha⟩
theorem actualMatrix_surjective_of_packetMap (n : ℕ)
(hpacket : Function.Surjective (packetMap d n)) :
Function.Surjective (actualMatrix d ((n : ℝ) * (scale d).radius)).mulVecLin := by
have h := actualMatrix_surjective_of_formalPackets d
((n : ℚ) * (scale d).radius) (fun P : WeightedPolynomials d n => P.val)
(fun P => by simpa only [Rat.cast_mul, Rat.cast_natCast] using P.property) hpacket
have hc : (((n : ℚ) * (scale d).radius : ℚ) : ℝ) =
(n : ℝ) * (scale d).radius := by push_cast; rfl
exact hc ▸ h
theorem cofinal_actualMatrix_of_eventual_packets
(hpacket : ∀ᶠ n : ℕ in atTop, Function.Surjective (packetMap d n))
(L : ℝ) : ∃ H : ℝ, L ≤ H ∧ Function.Surjective (actualMatrix d H).mulVecLin := by
obtain ⟨N,hN⟩ := eventually_atTop.mp hpacket
let n := max N (Nat.ceil (L / ((scale d).radius : ℝ)))
have hR : (0 : ℝ) < (scale d).radius := by exact_mod_cast (scale d).radius_pos
have hL : L ≤ (n : ℝ) * (scale d).radius := by
apply (div_le_iff₀ hR).mp
exact (Nat.le_ceil _).trans (by exact_mod_cast (le_max_right N
(Nat.ceil (L / ((scale d).radius : ℝ)))))
exact ⟨(n : ℝ) * (scale d).radius, hL,
actualMatrix_surjective_of_packetMap d n (hN n (le_max_left _ _))⟩
theorem globalInterpolation_of_eventual_packets
(hpacket : ∀ ν : ℝ, 2 < ν → ∀ d : FixedData ν,
∀ᶠ n : ℕ in atTop, Function.Surjective (packetMap d n)) :
GlobalInterpolationStatement := by
intro ν hν d L
exact cofinal_actualMatrix_of_eventual_packets d (hpacket ν hν d) L
end PiExponent.AdmissibleMatrixInterpolation
end
end OAI
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