| import OAI.NumberTheory.PiExponent.Polynomials.AuxiliaryPolynomial |
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| namespace OAI |
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| open scoped BigOperators |
| open Filter Topology |
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| namespace PiExponent |
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| noncomputable def literalJetSubstitution {m : ℕ} (r : Fin m → ℂ) |
| (G : Fin m → Polynomial ℂ) (j : ℕ) : |
| PiExponentApprox.FramePolynomial m →ₐ[ℂ] MvPolynomial (Fin m) (Polynomial ℂ) := |
| MvPolynomial.aeval (Fin.cases (MvPolynomial.C (1 + Polynomial.X)) |
| (fun i => MvPolynomial.C (Polynomial.C ((j : ℂ) * r i) + G i) + MvPolynomial.X i)) |
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| noncomputable def literalJetEvaluation {m : ℕ} (K : ℕ) |
| (V : Fin (m + 1) → ℝ) (H : ℝ) (r : Fin m → ℂ) |
| (G : Fin m → Polynomial ℂ) : |
| PiExponentApprox.FramePolynomial m →ₗ[ℂ] |
| ((Fin K × ↥(strictWeightedSimplex V H)) → ℂ) := |
| LinearMap.pi (fun ρ => (Polynomial.lcoeff ℂ (ρ.2.val 0)).comp |
| (((MvPolynomial.lcoeff (Polynomial ℂ) |
| (InterpolationMatrix.exponentVector (fun i => ρ.2.val i.succ))).restrictScalars ℂ).comp |
| (literalJetSubstitution r G ρ.1.val).toLinearMap)) |
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| @[simp] theorem literalJetEvaluation_apply {m : ℕ} (K : ℕ) |
| (V : Fin (m + 1) → ℝ) (H : ℝ) (r : Fin m → ℂ) |
| (G : Fin m → Polynomial ℂ) (p : PiExponentApprox.FramePolynomial m) |
| (ρ : Fin K × ↥(strictWeightedSimplex V H)) : |
| literalJetEvaluation K V H r G p ρ = |
| ((literalJetSubstitution r G ρ.1.val p).coeff |
| (InterpolationMatrix.exponentVector (fun i => ρ.2.val i.succ))).coeff (ρ.2.val 0) := rfl |
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| theorem literalJetSubstitution_monomial {m : ℕ} (r : Fin m → ℂ) |
| (G : Fin m → Polynomial ℂ) (j : ℕ) (e : Fin (m + 1) → ℕ) (z : ℂ) : |
| literalJetSubstitution r G j |
| (MvPolynomial.monomial (InterpolationMatrix.exponentVector e) z) = |
| MvPolynomial.C (Polynomial.C z) * |
| InterpolationMatrix.monomialImage r G j (e 0) (fun i => e i.succ) := by |
| classical |
| rw [literalJetSubstitution, MvPolynomial.aeval_monomial, |
| Finsupp.prod_fintype _ _ (fun _ => pow_zero _), Fin.prod_univ_succ] |
| simp only [InterpolationMatrix.exponentVector_apply, Fin.cases_zero, Fin.cases_succ, |
| InterpolationMatrix.monomialImage, ← map_pow] |
| rfl |
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| theorem literalJetSubstitution_polynomialOfCoefficients {m : ℕ} |
| (S : Finset (Fin (m + 1) → ℕ)) (x : S → ℂ) |
| (r : Fin m → ℂ) (G : Fin m → Polynomial ℂ) (j : ℕ) : |
| literalJetSubstitution r G j (polynomialOfCoefficients S x) = |
| InterpolationMatrix.sectionImage r G (fun c : S => c.val 0) |
| (fun c i => c.val i.succ) x j := by |
| classical |
| change literalJetSubstitution r G j |
| (∑ exponent : S, MvPolynomial.monomial |
| (InterpolationMatrix.exponentVector exponent.val) (x exponent)) = _ |
| simp only [map_sum, literalJetSubstitution_monomial, InterpolationMatrix.sectionImage] |
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| theorem eventually_exists_literalJet_auxiliaryPolynomial {m : ℕ} |
| (W V : Fin (m + 1) → ℚ) (hW : ∀ i, 0 < W i) (hV : ∀ i, 0 < V i) |
| (K : ℕ) {a : ℝ} (ha : 0 < a) |
| (hvol : (K : ℝ) * a ^ (m + 1) * (∏ i, (W i : ℝ)) / (∏ i, (V i : ℝ)) < 1) |
| (r : Fin m → ℂ) (G : Fin m → Polynomial ℂ) : |
| ∀ᶠ N : ℝ in atTop, ∃ p : PiExponentApprox.FramePolynomial m, p ≠ 0 ∧ |
| PiExponentApprox.HasWeightedDegreeLE (fun i => (W i : ℝ)) N p ∧ |
| ∀ ρ : Fin K × ↥(strictWeightedSimplex (fun i => (V i : ℝ)) (a * N)), |
| ((literalJetSubstitution r G ρ.1.val p).coeff |
| (InterpolationMatrix.exponentVector (fun i => ρ.2.val i.succ))).coeff (ρ.2.val 0) = 0 := by |
| filter_upwards [eventually_exists_auxiliaryPolynomial W V hW hV K ha hvol] with N hN |
| obtain ⟨p, hp, hw, he⟩ := hN (literalJetEvaluation K (fun i => (V i : ℝ)) (a * N) r G) |
| refine ⟨p, hp, hw, ?_⟩ |
| intro ρ |
| have h := congrFun he ρ |
| simpa only [literalJetEvaluation_apply, Pi.zero_apply] using h |
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| end PiExponent |
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| end OAI |
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