File size: 2,724 Bytes
f89a930 | 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 | 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
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