| import Mathlib.Algebra.MvPolynomial.Eval |
| import Mathlib.Analysis.Analytic.IsolatedZeros |
| import Mathlib.Analysis.Analytic.Order |
| import Mathlib.Analysis.Complex.CauchyIntegral |
| import Mathlib.Analysis.Normed.Module.Convex |
| import Mathlib.Analysis.SpecialFunctions.Complex.Log |
| import Mathlib.Tactic.FunProp |
| import OAI.AlgebraicGeometry.PlaneCurves.Automorphic |
|
|
| namespace OAI |
|
|
| /-! |
| # Analytic factorization, punctured identities, and holomorphic divisibility |
| -/ |
|
|
| section |
|
|
| /-! Genuine holomorphic division used in the collision argument. No Rouché theorem |
| or assumption of holomorphic quotients appears in these statements. -/ |
|
|
| namespace Nagata.W19 |
|
|
| open scoped BigOperators Topology |
| open Filter Set |
|
|
| /-- The divided slope of an analytic function extends analytically at its center. -/ |
| theorem analyticOnNhd_dslope {U : Set ℂ} {h : ℂ → ℂ} |
| (hh : AnalyticOnNhd ℂ h U) (a : ℂ) (ha : a ∈ U) : |
| AnalyticOnNhd ℂ (dslope h a) U := by |
| intro z hz |
| by_cases hza : z = a |
| · subst z |
| obtain ⟨P, hP⟩ := hh a ha |
| exact ⟨P.fslope, hP.has_fpower_series_dslope_fslope⟩ |
| · have hquot : AnalyticAt ℂ (fun w => (h w - h a) / (w - a)) z := |
| ((hh z hz).sub analyticAt_const).div (analyticAt_id.sub analyticAt_const) |
| (sub_ne_zero.mpr hza) |
| apply hquot.congr |
| filter_upwards [isOpen_ne.mem_nhds hza] with w hw |
| simp [dslope_of_ne h hw, slope, div_eq_mul_inv, smul_eq_mul, mul_comm] |
|
|
| /-- Divide out one centered zero while preserving holomorphicity on an open set. -/ |
| theorem exists_analytic_centered_factor |
| {U : Set ℂ} (hU : IsOpen U) (h : ℂ → ℂ) |
| (hh : AnalyticOnNhd ℂ h U) (a : ℂ) (ha : a ∈ U) |
| (k : ℕ) (ho : (k : ℕ∞) ≤ analyticOrderAt h a) : |
| ∃ g : ℂ → ℂ, AnalyticOnNhd ℂ g U ∧ |
| ∀ z, h z = (z - a) ^ k * g z := by |
| induction k generalizing h with |
| | zero => exact ⟨h, hh, fun z => by simp⟩ |
| | succ k ih => |
| have hzero : h a = 0 := by |
| apply apply_eq_zero_of_analyticOrderAt_ne_zero |
| intro hz |
| rw [hz] at ho |
| simp at ho |
| let g := dslope h a |
| have hg : AnalyticOnNhd ℂ g U := analyticOnNhd_dslope hh a ha |
| have heq : ∀ z, h z = (z - a) * g z := by |
| intro z |
| simpa [g, hzero, smul_eq_mul] using (sub_smul_dslope h a z).symm |
| have hfun : h = (fun z => z - a) * g := funext heq |
| have hlin : analyticOrderAt (fun z : ℂ => z - a) a = 1 := by |
| simp |
| have horder : analyticOrderAt h a = 1 + analyticOrderAt g a := by |
| rw [hfun, analyticOrderAt_mul (by fun_prop) |
| (hg.differentiableOn.analyticAt (hU.mem_nhds ha)), hlin] |
| have hgo : (k : ℕ∞) ≤ analyticOrderAt g a := by |
| rw [horder] at ho |
| apply (ENat.add_le_add_iff_left (by simp : (1 : ℕ∞) ≠ ⊤)).mp |
| simpa [Nat.cast_add, add_comm] using ho |
| obtain ⟨G, hG, heG⟩ := ih g hg hgo |
| refine ⟨G, hG, fun z => ?_⟩ |
| rw [heq z, heG z, pow_succ] |
| ac_rfl |
|
|
| /-- A factor nonvanishing at a point contributes zero to analytic vanishing order. -/ |
| theorem analytic_order_of_nonvanishing_analytic_factor |
| {U : Set ℂ} (hU : IsOpen U) {h g : ℂ → ℂ} |
| (hg : AnalyticOnNhd ℂ g U) {a b : ℂ} (hb : b ∈ U) |
| (hab : b ≠ a) (k : ℕ) (heq : ∀ z, h z = (z - a) ^ k * g z) : |
| analyticOrderAt h b = analyticOrderAt g b := by |
| have hfun : h = (fun z => (z - a) ^ k) * g := funext heq |
| have hfac : AnalyticAt ℂ (fun z => (z - a) ^ k) b := by fun_prop |
| rw [hfun, analyticOrderAt_mul hfac (hg.differentiableOn.analyticAt (hU.mem_nhds hb))] |
| have hz : analyticOrderAt (fun z => (z - a) ^ k) b = 0 := |
| hfac.analyticOrderAt_eq_zero.mpr (pow_ne_zero k (sub_ne_zero.mpr hab)) |
| rw [hz, zero_add] |
|
|
| /-- Simultaneously divide out finitely many distinct centers with prescribed order. -/ |
| theorem exists_analytic_finset_factorization |
| {I : Type*} (s : Finset I) {U : Set ℂ} (hU : IsOpen U) |
| (h : ℂ → ℂ) (hh : AnalyticOnNhd ℂ h U) |
| (ξ : I → ℂ) (hξ : Function.Injective ξ) (hmem : ∀ i ∈ s, ξ i ∈ U) |
| (k : ℕ) (ho : ∀ i ∈ s, (k : ℕ∞) ≤ analyticOrderAt h (ξ i)) : |
| ∃ g : ℂ → ℂ, AnalyticOnNhd ℂ g U ∧ |
| ∀ z, h z = (∏ i ∈ s, (z - ξ i) ^ k) * g z := by |
| classical |
| induction s using Finset.induction_on generalizing h with |
| | empty => exact ⟨h, hh, fun z => by simp⟩ |
| | @insert i s his ih => |
| obtain ⟨g, hg, hfg⟩ := exists_analytic_centered_factor hU h hh (ξ i) |
| (hmem i (Finset.mem_insert_self i s)) k (ho i (Finset.mem_insert_self i s)) |
| have hmems : ∀ j ∈ s, ξ j ∈ U := fun j hj => |
| hmem j (Finset.mem_insert_of_mem hj) |
| have hgos : ∀ j ∈ s, (k : ℕ∞) ≤ analyticOrderAt g (ξ j) := by |
| intro j hj |
| have hji : ξ j ≠ ξ i := hξ.ne (fun e => his (e ▸ hj)) |
| rw [← analytic_order_of_nonvanishing_analytic_factor hU hg (hmems j hj) hji k hfg] |
| exact ho j (Finset.mem_insert_of_mem hj) |
| obtain ⟨G, hG, heG⟩ := ih g hg hmems hgos |
| refine ⟨G, hG, fun z => ?_⟩ |
| rw [hfg z, heG z, Finset.prod_insert his, mul_assoc] |
|
|
| /-- Interface consumed by the holomorphic collision estimate. -/ |
| theorem exists_analytic_factorization |
| {q : ℕ} {U : Set ℂ} (hU : IsOpen U) (h : ℂ → ℂ) |
| (hh : AnalyticOnNhd ℂ h U) (ξ : Fin q → ℂ) |
| (hξ : Function.Injective ξ) (hmem : ∀ i, ξ i ∈ U) |
| (k : ℕ) (ho : ∀ i, (k : ℕ∞) ≤ analyticOrderAt h (ξ i)) : |
| ∃ g : ℂ → ℂ, AnalyticOnNhd ℂ g U ∧ |
| ∀ z, h z = (∏ i, (z - ξ i) ^ k) * g z := by |
| exact exists_analytic_finset_factorization Finset.univ hU h hh ξ hξ |
| (fun i _ => hmem i) k (fun i _ => ho i) |
|
|
| end Nagata.W19 |
|
|
| end |
|
|
| section |
|
|
| /-! Identity-principle zero-product results on the actual punctured complex |
| plane. Connectedness and nonemptiness are proved for that precise domain. -/ |
|
|
| namespace Nagata.W19 |
|
|
| theorem puncturedPlane_nonempty : ({z : ℂ | z ≠ 0} : Set ℂ).Nonempty := |
| ⟨1, one_ne_zero⟩ |
|
|
| theorem puncturedPlane_eq_exp_range : {z : ℂ | z ≠ 0} = Set.range Complex.exp := by |
| ext z |
| constructor |
| · intro hz |
| exact ⟨Complex.log z, Complex.exp_log hz⟩ |
| · rintro ⟨w, rfl⟩ |
| exact Complex.exp_ne_zero w |
|
|
| theorem puncturedPlane_isConnected : IsConnected {z : ℂ | z ≠ 0} := by |
| rw [puncturedPlane_eq_exp_range] |
| exact isConnected_range Complex.continuous_exp |
|
|
| theorem holomorphic_punctured_mul_zero (f g : ℂ → ℂ) |
| (hf : DifferentiableOn ℂ f {z : ℂ | z ≠ 0}) |
| (hg : DifferentiableOn ℂ g {z : ℂ | z ≠ 0}) |
| (hfg : ∀ z, z ≠ 0 → f z * g z = 0) : |
| (∀ z, z ≠ 0 → f z = 0) ∨ (∀ z, z ≠ 0 → g z = 0) := by |
| have ho : IsOpen {z : ℂ | z ≠ 0} := isOpen_ne |
| exact (hf.analyticOnNhd ho).eq_zero_or_eq_zero_of_mul_eq_zero |
| (hg.analyticOnNhd ho) hfg puncturedPlane_isConnected.isPreconnected |
|
|
| end Nagata.W19 |
|
|
| end |
|
|
| section |
|
|
| /-! Holomorphic wrappers for the checked analytic factorization core. -/ |
| namespace Nagata.W19 |
| open scoped BigOperators |
|
|
| theorem exists_holomorphic_centered_factor |
| {U : Set ℂ} (hU : IsOpen U) (h : ℂ → ℂ) |
| (hh : DifferentiableOn ℂ h U) (a : ℂ) (ha : a ∈ U) |
| (k : ℕ) (ho : (k : ℕ∞) ≤ analyticOrderAt h a) : |
| ∃ g : ℂ → ℂ, DifferentiableOn ℂ g U ∧ |
| ∀ z, h z = (z - a) ^ k * g z := by |
| obtain ⟨g, hg, heq⟩ := exists_analytic_centered_factor hU h (hh.analyticOnNhd hU) a ha k ho |
| exact ⟨g, hg.differentiableOn, heq⟩ |
|
|
| theorem analytic_order_of_nonvanishing_factor |
| {U : Set ℂ} (hU : IsOpen U) {h g : ℂ → ℂ} |
| (hg : DifferentiableOn ℂ g U) {a b : ℂ} (hb : b ∈ U) |
| (hab : b ≠ a) (k : ℕ) (heq : ∀ z, h z = (z - a) ^ k * g z) : |
| analyticOrderAt h b = analyticOrderAt g b := |
| analytic_order_of_nonvanishing_analytic_factor hU (hg.analyticOnNhd hU) hb hab k heq |
|
|
| theorem exists_holomorphic_finset_factorization |
| {I : Type*} (s : Finset I) {U : Set ℂ} (hU : IsOpen U) |
| (h : ℂ → ℂ) (hh : DifferentiableOn ℂ h U) |
| (ξ : I → ℂ) (hξ : Function.Injective ξ) (hmem : ∀ i ∈ s, ξ i ∈ U) |
| (k : ℕ) (ho : ∀ i ∈ s, (k : ℕ∞) ≤ analyticOrderAt h (ξ i)) : |
| ∃ g : ℂ → ℂ, DifferentiableOn ℂ g U ∧ |
| ∀ z, h z = (∏ i ∈ s, (z - ξ i) ^ k) * g z := by |
| obtain ⟨g, hg, heq⟩ := exists_analytic_finset_factorization s hU h |
| (hh.analyticOnNhd hU) ξ hξ hmem k ho |
| exact ⟨g, hg.differentiableOn, heq⟩ |
|
|
| /-- Interface consumed by the holomorphic collision estimate. -/ |
| theorem exists_holomorphic_factorization |
| {q : ℕ} {U : Set ℂ} (hU : IsOpen U) (h : ℂ → ℂ) |
| (hh : DifferentiableOn ℂ h U) (ξ : Fin q → ℂ) |
| (hξ : Function.Injective ξ) (hmem : ∀ i, ξ i ∈ U) |
| (k : ℕ) (ho : ∀ i, (k : ℕ∞) ≤ analyticOrderAt h (ξ i)) : |
| ∃ g : ℂ → ℂ, DifferentiableOn ℂ g U ∧ |
| ∀ z, h z = (∏ i, (z - ξ i) ^ k) * g z := by |
| obtain ⟨g, hg, heq⟩ := exists_analytic_factorization hU h |
| (hh.analyticOnNhd hU) ξ hξ hmem k ho |
| exact ⟨g, hg.differentiableOn, heq⟩ |
|
|
| end Nagata.W19 |
|
|
| end |
|
|
| section |
|
|
| /-! Polynomial evaluations in actual holomorphic section-cover functions. |
| The resulting zero-product alternative is the analytic input for primality |
| of the polynomial relation ideal; no function-domain property is assumed. -/ |
|
|
| namespace Nagata.W19 |
|
|
| theorem differentiableOn_polynomial_eval {σ : Type*} {U : Set ℂ} |
| (s : σ → ℂ → ℂ) (hs : ∀ i, DifferentiableOn ℂ (s i) U) |
| (P : MvPolynomial σ ℂ) : |
| DifferentiableOn ℂ (fun z => MvPolynomial.eval (fun i => s i z) P) U := by |
| induction P using MvPolynomial.induction_on with |
| | C a => simpa only [MvPolynomial.eval_C] using differentiableOn_const a |
| | add P Q hP hQ => |
| simp only [map_add] |
| exact hP.add hQ |
| | mul_X P i hP => |
| simp only [map_mul, MvPolynomial.eval_X] |
| exact hP.mul (hs i) |
|
|
| theorem polynomial_eval_punctured_mul_zero {σ : Type*} |
| (s : σ → ℂ → ℂ) (hs : ∀ i, DifferentiableOn ℂ (s i) {z : ℂ | z ≠ 0}) |
| (P Q : MvPolynomial σ ℂ) |
| (hPQ : ∀ z, z ≠ 0 → MvPolynomial.eval (fun i => s i z) (P * Q) = 0) : |
| (∀ z, z ≠ 0 → MvPolynomial.eval (fun i => s i z) P = 0) ∨ |
| (∀ z, z ≠ 0 → MvPolynomial.eval (fun i => s i z) Q = 0) := by |
| apply holomorphic_punctured_mul_zero |
| (fun z => MvPolynomial.eval (fun i => s i z) P) |
| (fun z => MvPolynomial.eval (fun i => s i z) Q) |
| (differentiableOn_polynomial_eval s hs P) (differentiableOn_polynomial_eval s hs Q) |
| intro z hz |
| simpa only [map_mul] using hPQ z hz |
|
|
| /-- The exact three-section specialization used for the degree-three map. -/ |
| theorem cubic_sections_polynomial_mul_zero {τ γ : ℂ} |
| (s : Fin 3 → Nagata.W08.automorphicSections τ 3 γ) |
| (P Q : MvPolynomial (Fin 3) ℂ) |
| (hPQ : ∀ z, z ≠ 0 → MvPolynomial.eval (fun i => (s i).val z) (P * Q) = 0) : |
| (∀ z, z ≠ 0 → MvPolynomial.eval (fun i => (s i).val z) P = 0) ∨ |
| (∀ z, z ≠ 0 → MvPolynomial.eval (fun i => (s i).val z) Q = 0) := by |
| exact polynomial_eval_punctured_mul_zero (fun i => (s i).val) |
| (fun i z hz => ((s i).property.2.1 z hz).differentiableWithinAt) P Q hPQ |
|
|
| end Nagata.W19 |
|
|
| end |
|
|
| end OAI |
|
|
