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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 | import OAI.NumberTheory.PiExponent.Approximation.WeightedSliceDegree
import OAI.NumberTheory.PiExponent.Geometry.CurveContact
namespace OAI
namespace PiExponent.FrameEquationFamily
open scoped BigOperators
open PiExponentApprox
abbrev BoundedWords {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ) :=
{word : List (Fin (m + 1)) // frameWordCost cost word ≤ bound}
noncomputable def equations {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ)
(F : FramePolynomial m) : BoundedWords cost bound → FramePolynomial m :=
fun word => polynomialFrameWord m word.1 F
theorem equations_mem {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ)
(F : FramePolynomial m) (word : BoundedWords cost bound) :
equations cost bound F word ∈ frameDerivativeIdeal cost bound F :=
frameWord_mem_derivativeIdeal cost bound F word.1 word.2
theorem span_equations {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ)
(F : FramePolynomial m) :
Ideal.span (Set.range (equations cost bound F)) =
frameDerivativeIdeal cost bound F := by
unfold frameDerivativeIdeal
congr 1
ext g
constructor
· rintro ⟨word, rfl⟩
exact ⟨word.1, word.2, rfl⟩
· rintro ⟨word, hword, rfl⟩
exact ⟨⟨word, hword⟩, rfl⟩
theorem monomialWeight_eq_weight {m : ℕ} (W : Fin (m + 1) → ℝ)
(d : Fin (m + 1) →₀ ℕ) :
monomialWeight W d = Finsupp.weight W d := by
simp only [monomialWeight, Finsupp.weight_eq_sum, nsmul_eq_mul]
theorem hasWeightedDegreeLE_iff_supportBound {m : ℕ} (W : Fin (m + 1) → ℝ)
(N : ℝ) (F : FramePolynomial m) :
HasWeightedDegreeLE W N F ↔ WeightedSliceDegree.SupportBound W N F := by
simp only [HasWeightedDegreeLE, WeightedSliceDegree.SupportBound, monomialWeight_eq_weight]
theorem equation_hasWeightedDegreeLE {m : ℕ} (cost W : Fin (m + 1) → ℝ)
(bound N : ℝ) (F : FramePolynomial m)
(hF : HasWeightedDegreeLE W N F) (hW : ∀ i, 0 ≤ W i)
(word : BoundedWords cost bound) :
HasWeightedDegreeLE W N (equations cost bound F word) :=
hF.polynomialFrameWord hW word.1
theorem equation_supportBound {m : ℕ} (cost W : Fin (m + 1) → ℝ)
(bound N : ℝ) (F : FramePolynomial m)
(hF : HasWeightedDegreeLE W N F) (hW : ∀ i, 0 ≤ W i)
(word : BoundedWords cost bound) :
WeightedSliceDegree.SupportBound W N (equations cost bound F word) :=
(hasWeightedDegreeLE_iff_supportBound W N _).mp
(equation_hasWeightedDegreeLE cost W bound N F hF hW word)
variable {E : Type*} [Field E] [Algebra ℂ E]
noncomputable def sliceEquations {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ)
(F : FramePolynomial m) (A : Finset (Fin (m + 1))) (b : Fin (m + 1) → E) :
BoundedWords cost bound → MvPolynomial (Fin A.card) E :=
fun word => WeightedSliceDegree.enumeratedSliceMap A b (equations cost bound F word)
theorem span_sliceEquations {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ)
(F : FramePolynomial m) (A : Finset (Fin (m + 1))) (b : Fin (m + 1) → E) :
Ideal.span (Set.range (sliceEquations cost bound F A b)) =
(frameDerivativeIdeal cost bound F).map
(WeightedSliceDegree.enumeratedSliceMap A b).toRingHom := by
rw [← span_equations cost bound F, Ideal.map_span]
congr 1
ext g
constructor
· rintro ⟨word, rfl⟩
exact ⟨equations cost bound F word, ⟨word, rfl⟩, rfl⟩
· rintro ⟨p, ⟨word, rfl⟩, rfl⟩
exact ⟨word, rfl⟩
theorem sliceEquation_supportBound {m : ℕ} (cost W : Fin (m + 1) → ℝ)
(bound N : ℝ) (F : FramePolynomial m)
(hF : HasWeightedDegreeLE W N F) (hW : ∀ i, 0 ≤ W i)
(A : Finset (Fin (m + 1))) (b : Fin (m + 1) → E)
(word : BoundedWords cost bound) :
WeightedSliceDegree.SupportBound (fun j : Fin A.card => W (A.equivFin.symm j)) N
(sliceEquations cost bound F A b word) :=
WeightedSliceDegree.supportBound_enumeratedSliceMap A b W hW N
(equations cost bound F word) (equation_supportBound cost W bound N F hF hW word)
theorem sliceEquation_weighted_sum_le {m : ℕ} (cost W : Fin (m + 1) → ℝ)
(bound N : ℝ) (F : FramePolynomial m)
(hF : HasWeightedDegreeLE W N F) (hW : ∀ i, 0 ≤ W i)
(A : Finset (Fin (m + 1))) (b : Fin (m + 1) → E)
(word : BoundedWords cost bound) :
∀ d ∈ (sliceEquations cost bound F A b word).support,
(∑ j ∈ d.support, (d j : ℝ) * W (A.equivFin.symm j)) ≤ N := by
have h := sliceEquation_supportBound cost W bound N F hF hW A b word
simpa only [WeightedSliceDegree.SupportBound, Finsupp.weight_apply, Finsupp.sum,
nsmul_eq_mul] using h
theorem sliceEquation_eval {m : ℕ} (cost : Fin (m + 1) → ℝ) (bound : ℝ)
(F : FramePolynomial m) (A : Finset (Fin (m + 1))) (b : Fin (m + 1) → E)
(word : BoundedWords cost bound) :
MvPolynomial.aeval (fun j : Fin A.card => b (A.equivFin.symm j))
(sliceEquations cost bound F A b word) =
MvPolynomial.aeval b (equations cost bound F word) :=
AlgHom.congr_fun (WeightedSliceDegree.enumeratedSliceMap_eval (C := ℂ) A b)
(equations cost bound F word)
end PiExponent.FrameEquationFamily
end OAI
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