openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Analysis /RealCoefficientNormalization.lean
| import OAI.NumberTheory.Jacobsthal.Harmonic.ComplexSquareScaling | |
| namespace OAI | |
| namespace Erdos970 | |
| section | |
| open scoped ComplexConjugate | |
| namespace ErdosComplexCurveFactors | |
| open ErdosCriticalGeometry | |
| noncomputable def realCoefficients (F : MV ℂ) : MV ℝ := | |
| AddMonoidAlgebra.ofCoeff (Finsupp.mapRange Complex.re (by simp) (AddMonoidAlgebra.coeff F)) | |
| theorem coeff_realCoefficients (F : MV ℂ) (m : Fin 2 →₀ ℕ) : | |
| (realCoefficients F).coeff m = (F.coeff m).re := rfl | |
| theorem complexify_realCoefficients (F : MV ℂ) (hF : conjugate F = F) : | |
| complexify (realCoefficients F) = F := by | |
| ext m | |
| change (MvPolynomial.map (algebraMap ℝ ℂ) (realCoefficients F)).coeff m = F.coeff m | |
| rw [MvPolynomial.coeff_map,coeff_realCoefficients] | |
| apply Complex.conj_eq_iff_re.mp | |
| have h := congrArg (fun P : MV ℂ => P.coeff m) hF | |
| simpa only [coeff_conjugate] using h | |
| def RealScalable (F : MV ℂ) : Prop := | |
| ∃ a : ℂ, a ≠ 0 ∧ ∃ R : MV ℝ, F = MvPolynomial.C a * complexify R | |
| theorem normalized_conjugate_eq (F : MV ℂ) (c : ℂ) (hc : c ≠ 0) | |
| (hrel : conjugate F = MvPolynomial.C c * F) (m : Fin 2 →₀ ℕ) | |
| (hm : F.coeff m ≠ 0) : | |
| conjugate (MvPolynomial.C (F.coeff m)⁻¹ * F) = MvPolynomial.C (F.coeff m)⁻¹ * F := by | |
| have hcoeff (k : Fin 2 →₀ ℕ) : conj (F.coeff k) = c*F.coeff k := by | |
| have h := congrArg (fun P : MV ℂ => P.coeff k) hrel | |
| simpa only [coeff_conjugate,MvPolynomial.coeff_C_mul] using h | |
| ext k | |
| simp only [coeff_conjugate,MvPolynomial.coeff_C_mul,map_mul,map_inv₀,hcoeff] | |
| field_simp [hc,hm] | |
| theorem realScalable_of_dvd_conjugate (F : MV ℂ) (hF : Irreducible F) | |
| (hdiv : F ∣ conjugate F) : RealScalable F := by | |
| obtain ⟨H,hH⟩ := hdiv | |
| have hu : IsUnit H := ((conjugate_irreducible hF).2 hH).resolve_left hF.not_isUnit | |
| obtain ⟨c,hc,hC⟩ := MvPolynomial.isUnit_iff_eq_C_of_isReduced.mp hu | |
| have hc0 : c ≠ 0 := hc.ne_zero | |
| have hrel : conjugate F = MvPolynomial.C c * F := by rw [hH,hC,mul_comm] | |
| obtain ⟨m,hm⟩ := MvPolynomial.exists_coeff_ne_zero hF.ne_zero | |
| let N := MvPolynomial.C (F.coeff m)⁻¹ * F | |
| have hN : conjugate N = N := normalized_conjugate_eq F c hc0 hrel m hm | |
| have hR : complexify (realCoefficients N) = N := complexify_realCoefficients N hN | |
| refine ⟨F.coeff m,hm,realCoefficients N,?_⟩ | |
| rw [hR] | |
| change F = MvPolynomial.C (F.coeff m) * (MvPolynomial.C (F.coeff m)⁻¹ * F) | |
| rw [← mul_assoc,← map_mul,mul_inv_cancel₀ hm,map_one,one_mul] | |
| theorem nonreal_not_dvd_conjugate (F : MV ℂ) (hF : Irreducible F) | |
| (hreal : ¬RealScalable F) : ¬F ∣ conjugate F := | |
| fun h => hreal (realScalable_of_dvd_conjugate F hF h) | |
| end ErdosComplexCurveFactors | |
| end | |
| end Erdos970 | |
| end OAI | |