| 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 |
|
|
