import Mathlib 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}