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import OAI.Algebra.FiniteTensor.MinorReduction
namespace OAI
/-! Algebraic derivatives, minor induction and polynomial Artin approximation. -/
noncomputable section
open scoped BigOperators
namespace PD4Tensor.Spreading
noncomputable section
variable {K σ : Type*} [Field K] [CharZero K]
theorem aeval_isAlgebraic {R S : Type*} [CommRing R] [IsDomain R] [CommRing S]
[Algebra R S] {f : S} (hf : IsAlgebraic R f) (P : Polynomial R) :
IsAlgebraic R (Polynomial.aeval f P) :=
Algebra.isAlgebraic_adjoin_singleton_iff.mpr hf _ (Polynomial.aeval_mem_adjoin_singleton R f)
theorem algebraic_series_pderiv (i : σ) (f : MvPowerSeries σ K)
(hf : IsAlgebraic (MvPolynomial σ K) f) :
IsAlgebraic (MvPolynomial σ K) (MvPowerSeries.pderiv i f) := by
classical
have hi : Function.Injective (algebraMap (MvPolynomial σ K) (MvPowerSeries σ K)) := by
intro x y h
apply MvPolynomial.coe_injective σ K
simpa only [MvPowerSeries.algebraMap_apply',Algebra.algebraMap_self,
MvPowerSeries.map_id,RingHom.id_apply] using h
obtain ⟨P,_,hP,hD⟩ := exists_relation_derivative_ne_zero hi f hf
have herr : ∀ Q : Polynomial (MvPolynomial σ K),
IsAlgebraic (MvPolynomial σ K)
(MvPowerSeries.pderiv i (Polynomial.aeval f Q)-
Polynomial.aeval f Q.derivative*MvPowerSeries.pderiv i f) := by
intro Q
induction Q using Polynomial.induction_on' with
| add Q R hQ hR =>
have heq : MvPowerSeries.pderiv i (Polynomial.aeval f (Q+R)) -
Polynomial.aeval f (Q+R).derivative * MvPowerSeries.pderiv i f =
(MvPowerSeries.pderiv i (Polynomial.aeval f Q) -
Polynomial.aeval f Q.derivative * MvPowerSeries.pderiv i f) +
(MvPowerSeries.pderiv i (Polynomial.aeval f R) -
Polynomial.aeval f R.derivative * MvPowerSeries.pderiv i f) := by
simp only [map_add,add_mul]
ring
rw [heq]
exact hQ.add hR
| monomial j a =>
have ha : IsAlgebraic (MvPolynomial σ K)
(MvPowerSeries.pderiv i (algebraMap (MvPolynomial σ K) (MvPowerSeries σ K) a)) := by
simpa only [MvPowerSeries.algebraMap_apply',Algebra.algebraMap_self,
MvPowerSeries.map_id,RingHom.id_apply,MvPowerSeries.pderiv_coe] using
(isAlgebraic_algebraMap (R:=MvPolynomial σ K) (A:=MvPowerSeries σ K)
(MvPolynomial.pderiv i a))
have heq : MvPowerSeries.pderiv i (Polynomial.aeval f (Polynomial.monomial j a)) -
Polynomial.aeval f (Polynomial.monomial j a).derivative * MvPowerSeries.pderiv i f =
MvPowerSeries.pderiv i (algebraMap (MvPolynomial σ K) (MvPowerSeries σ K) a) * f^j := by
simp only [Polynomial.aeval_monomial,Polynomial.derivative_monomial,
Derivation.leibniz,Derivation.leibniz_pow,smul_eq_mul,map_mul,map_natCast,
nsmul_eq_mul]
ring
rw [heq]
exact ha.mul (hf.pow j)
have hprod : IsAlgebraic (MvPolynomial σ K)
(Polynomial.aeval f P.derivative*MvPowerSeries.pderiv i f) := by
simpa only [hP,map_zero,zero_sub,neg_neg] using (herr P).neg
exact IsAlgebraic.of_mul (mem_nonZeroDivisors_of_ne_zero hD)
(aeval_isAlgebraic hf _) hprod
end
end PD4Tensor.Spreading
namespace PD4Tensor.Spreading
noncomputable section
variable {K σ τ : Type*} [Field K] [Finite σ]
theorem algebraic_series_rename (j : σ → τ) (hj : Function.Injective j)
(f : MvPowerSeries σ K) (hf : IsAlgebraic (MvPolynomial σ K) f) :
IsAlgebraic (MvPolynomial τ K) (MvPowerSeries.rename j f) := by
apply hf.ringHom_of_comp_eq (MvPolynomial.rename j).toRingHom
(MvPowerSeries.rename j).toRingHom (MvPolynomial.rename_injective j hj)
apply RingHom.ext
intro p
simp only [RingHom.comp_apply,AlgHom.toRingHom_eq_coe,MvPowerSeries.algebraMap_apply',
Algebra.algebraMap_self,MvPowerSeries.map_id,RingHom.id_apply]
exact (MvPowerSeries.rename_coe j p).symm
end
end PD4Tensor.Spreading
namespace PD4Tensor.Spreading
noncomputable section
open MvPolynomial
variable {K τ : Type*} [Field K] [CharZero K] [Finite τ]
theorem aeval_minorPolynomial {R A σ ρ : Type*} [CommRing R] [CommRing A]
[Algebra R A] [Fintype σ] [DecidableEq σ]
(a : (σ ⊕ ρ) → A) (f : σ → MvPolynomial (σ ⊕ ρ) R) :
aeval a (minorPolynomial f)=Matrix.det (fun k i=>aeval a (pderiv (Sum.inl i) (f k))) := by
unfold minorPolynomial
exact RingHom.map_det (aeval a : MvPolynomial (σ ⊕ ρ) R →ₐ[R] A).toRingHom _
omit [CharZero K] [Finite τ] in
theorem rename_polynomial_square {υ : Type*} (e : τ ≃ υ) :
(algebraMap (MvPolynomial υ K) (MvPowerSeries υ K)).comp (MvPolynomial.rename e).toRingHom=
(MvPowerSeries.renameEquiv K e).toRingHom.comp
(algebraMap (MvPolynomial τ K) (MvPowerSeries τ K)) := by
apply MvPolynomial.ringHom_ext
· intro c
simp [MvPowerSeries.algebraMap_apply']
· intro i
simp [MvPowerSeries.algebraMap_apply']
theorem minor_artin_step
(ih : ∀ p : τ,PolynomialArtinAt K {i : τ // i≠p}) : MinorArtinAt K τ := by
classical
intro σ ρ _ _ _ _ f a ha hJ n
let δ := aeval a (minorPolynomial f)
have hδ : δ≠0 := by dsimp [δ]; rw [aeval_minorPolynomial]; exact hJ
by_cases hu : IsUnit δ
· let F := fun k=>map (algebraMap (MvPolynomial τ K) (MvPowerSeries τ K)) (f k)
have hev (v : (σ ⊕ ρ) → MvPowerSeries τ K) (q : MvPolynomial (σ ⊕ ρ) (MvPolynomial τ K)) :
eval v (map (algebraMap (MvPolynomial τ K) (MvPowerSeries τ K)) q)=aeval v q := by
rw [←eval₂_eq_eval_map,aeval_def]
have hFcoeff (k : σ) (d) : IsAlgebraic (MvPolynomial τ K) ((F k).coeff d) := by
rw [coeff_map]
exact isAlgebraic_algebraMap _
have hJF : IsUnit (Matrix.det (fun k i=>eval a (pderiv (Sum.inl i) (F k)))) := by
simp_rw [F,pderiv_map,hev]
rwa [←aeval_minorPolynomial]
obtain ⟨b,hb,hbalg,hba⟩ := unit_artin_step F hFcoeff a (fun k=>by rw [hev]; exact ha k) hJF n
exact ⟨b,fun k=>by rw [←hev]; exact hb k,hbalg,hba⟩
· have hzero : MvPowerSeries.constantCoeff δ=0 := by
rw [MvPowerSeries.isUnit_iff_constantCoeff,isUnit_iff_ne_zero,not_not] at hu
exact hu
obtain ⟨p,c,hp,hreg⟩ := exists_regular_linear_chart δ hδ hzero
let υ := {i : τ // i≠p}
let e := Equiv.optionSubtypeNe p
let C := linearSeriesChart p c hp
let E := C.trans (MvPowerSeries.renameEquiv K e.symm)
let P := (MvPolynomial.rename e.symm).toRingHom.comp (polynomialChart p c)
have hsq : (algebraMap (MvPolynomial (Option υ) K) (MvPowerSeries (Option υ) K)).comp P=
E.toRingHom.comp (algebraMap (MvPolynomial τ K) (MvPowerSeries τ K)) := by
apply RingHom.ext
intro q
change (algebraMap (MvPolynomial (Option υ) K) (MvPowerSeries (Option υ) K))
(MvPolynomial.rename e.symm (polynomialChart p c q))=
MvPowerSeries.rename e.symm (C ((algebraMap (MvPolynomial τ K) (MvPowerSeries τ K)) q))
have hr := RingHom.congr_fun (rename_polynomial_square (K:=K) e.symm) (polynomialChart p c q)
have hc := RingHom.congr_fun (polynomialChart_square p c hp) q
exact hr.trans (congrArg (MvPowerSeries.rename e.symm) hc)
let f' := fun k=>map P (f k)
let a' := fun i=>E (a i)
have hev (v : (σ ⊕ ρ) → MvPowerSeries τ K) (q : MvPolynomial (σ ⊕ ρ) (MvPolynomial τ K)) :
aeval (fun i=>E (v i)) (map P q)=E (aeval v q) :=
aeval_map_square P E.toRingHom hsq q v
have hminor : minorPolynomial f'=map P (minorPolynomial f) := (minorPolynomial_map P f).symm
obtain ⟨b,hb,hbalg,hba⟩ := option_artin_step (ih p) f' a'
(fun k=>by rw [hev,ha,map_zero])
(by
rw [hminor,hev]
change MvPowerSeries.subst (axisVars (K:=K) none)
(MvPowerSeries.rename e.symm (C δ))≠0
rwa [option_axis_rename])
(by
rw [hminor,hev]
exact fun h=>hu ((isUnit_map_iff E _).mp h)) n
refine ⟨fun i=>E.symm (b i),?_,?_,?_⟩
· intro k
apply E.injective
rw [map_zero,←hev]
simpa only [AlgEquiv.apply_symm_apply] using hb k
· intro i
change IsAlgebraic (MvPolynomial τ K) (C.symm (MvPowerSeries.rename e (b i)))
apply algebraic_linearSeriesChart_symm
exact algebraic_series_rename e e.injective _ (hbalg i)
· intro i
have h := series_equiv_jet E.symm.toRingEquiv (hba i)
change E.symm (b i-a' i)∈_ at h
simpa only [a',map_sub,AlgEquiv.symm_apply_apply] using h
end
end PD4Tensor.Spreading
namespace PD4Tensor.Spreading
noncomputable section
universe u v
variable (K : Type u) [Field K] [CharZero K]
theorem polynomial_artin (τ : Type v) [Finite τ] : PolynomialArtinAt K τ := by
classical
have h : ∀ n : ℕ,∀ (υ : Type v) [Finite υ],Nat.card υ=n → PolynomialArtinAt K υ := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro υ _ hn
apply polynomial_artin_of_minor
apply minor_artin_step
intro p
have hlt : Nat.card {i : υ // i≠p}<n := by
let := Fintype.ofFinite υ
let := Fintype.ofFinite {i : υ // i≠p}
rw [←hn,←Fintype.card_eq_nat_card,←Fintype.card_eq_nat_card]
exact Fintype.card_subtype_lt (x:=p) (by simp)
exact ih _ hlt _ rfl
exact h _ τ rfl
end
end PD4Tensor.Spreading
end
end OAI
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