| import OAI.AlgebraicGeometry.PlaneCurves.AnalyticTaylor |
| import OAI.AlgebraicGeometry.PlaneCurves.CoefficientSpaces |
| import OAI.AlgebraicGeometry.PlaneCurves.FrechetMultiplicity |
| import OAI.AlgebraicGeometry.PlaneCurves.MixedDerivatives |
| import OAI.AlgebraicGeometry.PlaneCurves.NormalCoefficients |
|
|
| namespace OAI |
|
|
| /-! |
| # Analytic coefficient order, nested order, and ideal-power multiplicity |
| -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.W18 |
| open Nagata.CoefficientSpaces |
|
|
| /-- Exact transverse derivative of an actual polynomial of function coefficients. |
| No analyticity assumption on the coefficient functions is needed for this identity. -/ |
| theorem scalarExpression_iteratedDeriv_zero |
| (F : Polynomial (ℂ → ℂ)) (j : ℕ) (z : ℂ) : |
| iteratedDeriv j (fun w => scalarExpression F z w) 0 = |
| (j.factorial : ℂ) * F.coeff j z := by |
| classical |
| simp only [scalarExpression, Polynomial.sum_def] |
| rw [iteratedDeriv_fun_sum (fun k _ => |
| contDiffAt_const.mul (contDiffAt_id.pow k))] |
| simp only [iteratedDeriv_const_mul_field, iteratedDeriv_fun_pow_zero] |
| rw [Finset.sum_eq_single j] |
| · simp [mul_comm] |
| · intro k hk hkj |
| simp [Ne.symm hkj] |
| · intro hj |
| have hc : F.coeff j = 0 := by |
| simpa only [Polynomial.mem_support_iff, not_not] using hj |
| simp [hc] |
|
|
| /-- Nonzero constant multiplication preserves analytic order of an analytic germ. -/ |
| theorem analyticOrderAt_nonzero_const_mul |
| (f : ℂ → ℂ) (a c : ℂ) (hc : c ≠ 0) (hf : AnalyticAt ℂ f a) : |
| analyticOrderAt (fun z => c * f z) a = analyticOrderAt f a := by |
| have hconst : AnalyticAt ℂ (fun _ : ℂ => c) a := analyticAt_const |
| change analyticOrderAt ((fun _ : ℂ => c) * f) a = _ |
| rw [analyticOrderAt_mul hconst hf, hconst.analyticOrderAt_eq_zero.mpr hc, zero_add] |
|
|
| /-- Joint analyticity forces analyticity of every actual function coefficient, |
| including indices above the polynomial degree. -/ |
| theorem scalar_coefficient_analyticAt |
| (F : Polynomial (ℂ → ℂ)) (a : ℂ) (j : ℕ) |
| (hF : AnalyticAt ℂ (fun p : ℂ × ℂ => scalarExpression F p.1 p.2) (a, 0)) : |
| AnalyticAt ℂ (F.coeff j) a := by |
| have hd := Nagata.W19.analyticAt_transverse_iteratedDeriv a hF j |
| have he : (fun x => iteratedDeriv j (fun y => scalarExpression F x y) 0) = |
| (fun x => (j.factorial : ℂ) * F.coeff j x) := by |
| funext x |
| exact scalarExpression_iteratedDeriv_zero F j x |
| rw [he] at hd |
| have hc : (j.factorial : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero j) |
| have hh := (analyticAt_const (v := (j.factorial : ℂ)⁻¹)).mul hd |
| change AnalyticAt ℂ (fun x => (j.factorial : ℂ)⁻¹ * |
| ((j.factorial : ℂ) * F.coeff j x)) a at hh |
| simpa only [← mul_assoc, inv_mul_cancel₀ hc, one_mul] using hh |
|
|
| /-- Ordinary analytic order at a zero-displacement point gives order `m-j` |
| of every genuine scalar coefficient, with natural truncated subtraction. -/ |
| theorem scalar_coefficient_order |
| (F : Polynomial (ℂ → ℂ)) (a : ℂ) (m j : ℕ) |
| (hF : Nagata.Workers.W28.HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun p : ℂ × ℂ => scalarExpression F p.1 p.2) (a, 0) m) : |
| ((m - j : ℕ) : ℕ∞) ≤ analyticOrderAt (F.coeff j) a := by |
| obtain ⟨P, hP, hzero⟩ := hF |
| have ho := Nagata.W19.transverse_iteratedDeriv_order_of_powerSeries a hP m j hzero |
| have he : (fun x => iteratedDeriv j (fun y => scalarExpression F x y) 0) = |
| (fun x => (j.factorial : ℂ) * F.coeff j x) := by |
| funext x |
| exact scalarExpression_iteratedDeriv_zero F j x |
| rw [he] at ho |
| rw [analyticOrderAt_nonzero_const_mul (F.coeff j) a (j.factorial : ℂ) |
| (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero j)) |
| (scalar_coefficient_analyticAt F a j ⟨P, hP⟩)] at ho |
| exact ho |
|
|
| end Nagata.W18 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! |
| # Actual analytic Taylor order implies ordinary polynomial multiplicity |
| -/ |
|
|
| noncomputable section |
| namespace Nagata.W18 |
|
|
| open Nagata.Workers.W30 |
| open Nagata.AffineMultiplicity |
| open Nagata.Workers.W28 |
|
|
| /-- Centered convergent-series vanishing forces every low polynomial coefficient |
| zero. The nonzero factorials are cancelled in the actual field ℂ. -/ |
| theorem polynomial_coeff_zero_of_centered_powerSeries |
| (P : MvPolynomial (Fin 2) ℂ) |
| (S : FormalMultilinearSeries ℂ (ℂ × ℂ) ℂ) |
| (hS : HasFPowerSeriesAt (fun z : ℂ × ℂ => planeValue P z.1 z.2) S 0) |
| (m : ℕ) (hzero : ∀ n < m, S n = 0) : |
| ∀ ell b : ℕ, ell + b < m → P.coeff (exponentPair ell b) = 0 := by |
| intro ell b hlt |
| have hd := Nagata.W19.mixed_iteratedDeriv_zero_of_powerSeries 0 hS m hzero ell b hlt |
| change mixedDerivative ell b P 0 0 = 0 at hd |
| rw [mixedDerivative_zero] at hd |
| have hfac : (ell.factorial : ℂ) * (b.factorial : ℂ) ≠ 0 := |
| mul_ne_zero (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero ell)) |
| (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero b)) |
| exact (mul_eq_zero.mp hd).resolve_left hfac |
|
|
| /-- A convergent series at any actual affine point with low terms zero forces |
| membership in the corresponding point-ideal power. -/ |
| theorem polynomial_ideal_order_of_powerSeries |
| (P : MvPolynomial (Fin 2) ℂ) (p : ℂ × ℂ) |
| (S : FormalMultilinearSeries ℂ (ℂ × ℂ) ℂ) |
| (hS : HasFPowerSeriesAt (fun z : ℂ × ℂ => planeValue P z.1 z.2) S p) |
| (m : ℕ) (hzero : ∀ n < m, S n = 0) : |
| orderAtLeast (planePoint p.1 p.2) m P := by |
| have hshift := hS.comp_sub (-p) |
| simp only [add_neg_cancel] at hshift |
| have htranslated : HasFPowerSeriesAt |
| (fun z : ℂ × ℂ => planeValue (translateHom (planePoint p.1 p.2) P) z.1 z.2) |
| S 0 := by |
| apply hshift.congr |
| exact Filter.Eventually.of_forall fun z => by |
| dsimp only |
| rw [planeValue_translate] |
| simp only [sub_neg_eq_add, Prod.fst_add, Prod.snd_add] |
| apply (orderAtLeast_finTwo_iff_taylor_coeff (planePoint p.1 p.2) m P).mpr |
| exact polynomial_coeff_zero_of_centered_powerSeries _ S htranslated m hzero |
|
|
| theorem planePoint_eq_coordinate_function (p : ℂ × ℂ) : |
| planePoint p.1 p.2 = (fun i : Fin 2 => if i = 0 then p.1 else p.2) := by |
| funext i |
| fin_cases i <;> rfl |
|
|
| theorem polynomial_analytic_order_implies_ideal_order |
| (P : MvPolynomial (Fin 2) ℂ) (p : ℂ × ℂ) (m : ℕ) |
| (hP : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => MvPolynomial.eval |
| (fun i => if i = 0 then z.1 else z.2) P) p m) : |
| orderAtLeast (fun i : Fin 2 => if i = 0 then p.1 else p.2) m P := by |
| obtain ⟨S, hS, hzero⟩ := hP |
| have hseries : HasFPowerSeriesAt (fun z : ℂ × ℂ => planeValue P z.1 z.2) S p := by |
| simpa only [planeValue, planePoint_eq_coordinate_function] using hS |
| have h := polynomial_ideal_order_of_powerSeries P p S hseries m hzero |
| rwa [planePoint_eq_coordinate_function] at h |
|
|
| /-- Ordinary ideal-power multiplicity and existence of a genuine convergent |
| Taylor expansion with low terms zero are equivalent for actual plane polynomials. -/ |
| theorem polynomial_analytic_order_iff_ideal_order |
| (P : MvPolynomial (Fin 2) ℂ) (p : ℂ × ℂ) (m : ℕ) : |
| HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => MvPolynomial.eval |
| (fun i => if i = 0 then z.1 else z.2) P) p m ↔ |
| orderAtLeast (fun i : Fin 2 => if i = 0 then p.1 else p.2) m P := |
| ⟨polynomial_analytic_order_implies_ideal_order P p m, |
| polynomial_order_implies_analytic_order P p m⟩ |
|
|
| /-- The analytic normal-limit order can now be passed to the actual nested |
| polynomial point ideal used by the coefficient and fiber-order package. -/ |
| theorem nested_ideal_order_of_analytic_order |
| (P : MvPolynomial (Fin 2) ℂ) (a c : ℂ) (m : ℕ) |
| (hP : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => MvPolynomial.eval |
| (fun i => if i = 0 then z.1 else z.2) P) (a, c) m) : |
| bivariateToNested P ∈ (nestedPointIdeal a c) ^ m := by |
| apply nested_order_of_affine_order a c m P |
| have h := polynomial_analytic_order_implies_ideal_order P (a, c) m hP |
| convert h using 1 |
| funext i |
| fin_cases i <;> rfl |
|
|
| /-- Analytic order directly forces base order of the highest possible fiber |
| coefficient of the actual local polynomial. -/ |
| theorem analytic_top_coefficient_base_divisibility |
| (P : MvPolynomial (Fin 2) ℂ) (a c : ℂ) (m J : ℕ) |
| (hP : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => MvPolynomial.eval |
| (fun i => if i = 0 then z.1 else z.2) P) (a, c) m) |
| (hJ : (bivariateToNested P).natDegree ≤ J) : |
| (Polynomial.X - Polynomial.C a) ^ (m - J) ∣ (bivariateToNested P).coeff J := |
| top_coefficient_base_divisibility _ a c m J hJ |
| (nested_ideal_order_of_analytic_order P a c m hP) |
|
|
| /-- Analytic order directly gives the actual marked fiber root divisibility. -/ |
| theorem analytic_fiber_power_divides |
| (P : MvPolynomial (Fin 2) ℂ) (a c : ℂ) (m : ℕ) |
| (hP : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => MvPolynomial.eval |
| (fun i => if i = 0 then z.1 else z.2) P) (a, c) m) : |
| (Polynomial.X - Polynomial.C c) ^ m ∣ fiberRestriction a (bivariateToNested P) := |
| fiber_power_divides_of_surface_order _ a c m |
| (nested_ideal_order_of_analytic_order P a c m hP) |
|
|
| end Nagata.W18 |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Direct analytic-to-ideal interface for the actual nested normal polynomial. |
| Evaluation below is ordinary polynomial evaluation in W followed by U. -/ |
|
|
| noncomputable section |
| namespace Nagata.W18 |
| open Nagata.Workers.W28 |
|
|
| /-- Explicit inverse-coordinate evaluation identity. -/ |
| theorem nestedEvaluation_eq_polynomialEvaluation |
| (H : Polynomial (Polynomial ℂ)) (z : ℂ × ℂ) : |
| (H.eval (Polynomial.C z.2)).eval z.1 = |
| MvPolynomial.eval (fun i : Fin 2 => if i = 0 then z.1 else z.2) |
| (nestedToBivariate H) := by |
| have h := eval_bivariateToNested z.1 z.2 (nestedToBivariate H) |
| have hinv : bivariateToNested (nestedToBivariate H) = H := |
| RingHom.congr_fun bivariateToNested_comp_nestedToBivariate H |
| rw [hinv] at h |
| have hcoords : ![z.1, z.2] = (fun i : Fin 2 => if i = 0 then z.1 else z.2) := by |
| funext i |
| fin_cases i <;> rfl |
| simpa only [hcoords] using h |
|
|
| /-- A genuine convergent analytic-order statement for H(U,W) implies its |
| actual nested point-ideal-power membership. -/ |
| theorem nested_polynomial_ideal_order_of_analytic_order |
| (H : Polynomial (Polynomial ℂ)) (a c : ℂ) (m : ℕ) |
| (hH : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => (H.eval (Polynomial.C z.2)).eval z.1) (a, c) m) : |
| H ∈ (nestedPointIdeal a c) ^ m := by |
| have hP : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => MvPolynomial.eval |
| (fun i => if i = 0 then z.1 else z.2) (nestedToBivariate H)) (a, c) m := by |
| simpa only [nestedEvaluation_eq_polynomialEvaluation] using hH |
| have h := nested_ideal_order_of_analytic_order (nestedToBivariate H) a c m hP |
| have hinv : bivariateToNested (nestedToBivariate H) = H := |
| RingHom.congr_fun bivariateToNested_comp_nestedToBivariate H |
| rwa [hinv] at h |
|
|
| /-- Direct genuine-series version of the nested polynomial order bridge. -/ |
| theorem nested_polynomial_ideal_order_of_powerSeries |
| (H : Polynomial (Polynomial ℂ)) (a c : ℂ) |
| (S : FormalMultilinearSeries ℂ (ℂ × ℂ) ℂ) |
| (hS : HasFPowerSeriesAt |
| (fun z : ℂ × ℂ => (H.eval (Polynomial.C z.2)).eval z.1) S (a, c)) |
| (m : ℕ) (hzero : ∀ n < m, S n = 0) : |
| H ∈ (nestedPointIdeal a c) ^ m := |
| nested_polynomial_ideal_order_of_analytic_order H a c m ⟨S, hS, hzero⟩ |
|
|
| /-- The analytic limit's actual local polynomial top coefficient has the |
| required base vanishing divisibility. -/ |
| theorem nested_analytic_top_coefficient_divisibility |
| (H : Polynomial (Polynomial ℂ)) (a c : ℂ) (m J : ℕ) |
| (hH : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => (H.eval (Polynomial.C z.2)).eval z.1) (a, c) m) |
| (hJ : H.natDegree ≤ J) : |
| (Polynomial.X - Polynomial.C a) ^ (m - J) ∣ H.coeff J := |
| top_coefficient_base_divisibility H a c m J hJ |
| (nested_polynomial_ideal_order_of_analytic_order H a c m hH) |
| |
| /-- The analytic limit's actual local polynomial has the required fiber root. -/ |
| theorem nested_analytic_fiber_divisibility |
| (H : Polynomial (Polynomial ℂ)) (a c : ℂ) (m : ℕ) |
| (hH : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun z : ℂ × ℂ => (H.eval (Polynomial.C z.2)).eval z.1) (a, c) m) : |
| (Polynomial.X - Polynomial.C c) ^ m ∣ fiberRestriction a H := |
| fiber_power_divides_of_surface_order H a c m |
| (nested_polynomial_ideal_order_of_analytic_order H a c m hH) |
|
|
| end Nagata.W18 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
