import Mathlib.Analysis.Analytic.Constructions import Mathlib.Analysis.Analytic.IsolatedZeros import Mathlib.Analysis.Complex.Basic import Nevanlinna.analyticAt noncomputable def AnalyticOn.order {f : ℂ → ℂ} {U : Set ℂ} (hf : AnalyticOn ℂ f U) : U → ℕ∞ := fun u ↦ (hf u u.2).order theorem AnalyticOn.order_eq_nat_iff {f : ℂ → ℂ} {U : Set ℂ} {z₀ : U} (hf : AnalyticOn ℂ f U) (n : ℕ) : hf.order z₀ = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by constructor -- Direction → intro hn obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ z₀.2) n).1 hn -- Define a candidate function; this is (f z) / (z - z₀) ^ n with the -- removable singularity removed let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n -- Describe g near z₀ have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by rw [eventually_nhds_iff] obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc use t constructor · intro y h₁y by_cases h₂y : y = z₀ · dsimp [g]; simp [h₂y] · dsimp [g]; simp [h₂y] rw [div_eq_iff_mul_eq, eq_comm, mul_comm] exact h₁t y h₁y norm_num rw [sub_eq_zero] tauto · constructor · assumption · assumption -- Describe g near points z₁ that are different from z₀ have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by intro hz₁ rw [eventually_nhds_iff] use {z₀.1}ᶜ constructor · intro y hy simp at hy simp [g, hy] · exact ⟨isOpen_compl_singleton, hz₁⟩ -- Use g and show that it has all required properties use g constructor · -- AnalyticOn ℂ g U intro z h₁z by_cases h₂z : z = z₀ · rw [h₂z] apply AnalyticAt.congr h₁gloc exact Filter.EventuallyEq.symm g_near_z₀ · simp_rw [eq_comm] at g_near_z₁ apply AnalyticAt.congr _ (g_near_z₁ h₂z) apply AnalyticAt.div exact hf z h₁z apply AnalyticAt.pow apply AnalyticAt.sub apply analyticAt_id apply analyticAt_const simp rw [sub_eq_zero] tauto · constructor · simp [g]; tauto · intro z by_cases h₂z : z = z₀ · rw [h₂z, g_near_z₀.self_of_nhds] exact h₃gloc.self_of_nhds · rw [(g_near_z₁ h₂z).self_of_nhds] simp [h₂z] rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel₀] simp; norm_num rw [sub_eq_zero] tauto -- direction ← intro h obtain ⟨g, h₁g, h₂g, h₃g⟩ := h dsimp [AnalyticOn.order] rw [AnalyticAt.order_eq_nat_iff] use g exact ⟨h₁g z₀ z₀.2, ⟨h₂g, Filter.Eventually.of_forall h₃g⟩⟩ theorem AnalyticOn.eliminateZeros {f : ℂ → ℂ} {U : Set ℂ} {A : Finset U} (hf : AnalyticOn ℂ f U) (n : ℂ → ℕ) : (∀ a ∈ A, hf.order a = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z) -- case empty simp use f simp exact hf -- case insert intro b₀ B hb iHyp intro hBinsert obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha)) have : (h₁g₀ b₀ b₀.2).order = n b₀ := by rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)] let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a.1) have : f = fun z ↦ φ z * g₀ z := by funext z rw [h₃g₀ z] rfl simp_rw [this] have h₁φ : AnalyticAt ℂ φ b₀ := by dsimp [φ] apply Finset.analyticAt_prod intro b _ apply AnalyticAt.pow apply AnalyticAt.sub apply analyticAt_id ℂ exact analyticAt_const have h₂φ : h₁φ.order = (0 : ℕ) := by rw [AnalyticAt.order_eq_nat_iff h₁φ 0] use φ constructor · assumption · constructor · dsimp [φ] push_neg rw [Finset.prod_ne_zero_iff] intro a ha simp have : ¬ (b₀.1 - a.1 = 0) := by by_contra C rw [sub_eq_zero] at C rw [SetCoe.ext C] at hb tauto tauto · simp rw [AnalyticAt.order_mul h₁φ (h₁g₀ b₀ b₀.2)] rw [h₂φ] simp obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ (n b₀)).1 this use g₁ constructor · exact h₁g₁ · constructor · intro a h₁a by_cases h₂a : a = b₀ · rwa [h₂a] · let A' := Finset.mem_of_mem_insert_of_ne h₁a h₂a let B' := h₃g₁ a let C' := h₂g₀ a A' rw [B'] at C' exact right_ne_zero_of_smul C' · intro z let A' := h₃g₀ z rw [h₃g₁ z] at A' rw [A'] rw [← smul_assoc] congr simp rw [Finset.prod_insert] ring exact hb theorem XX {f : ℂ → ℂ} {U : Set ℂ} (hU : IsPreconnected U) (h₁f : AnalyticOn ℂ f U) (h₂f : ∃ u ∈ U, f u ≠ 0) : ∀ (hu : u ∈ U), (h₁f u hu).order.toNat = (h₁f u hu).order := by intro hu apply ENat.coe_toNat by_contra C rw [(h₁f u hu).order_eq_top_iff] at C rw [← (h₁f u hu).frequently_zero_iff_eventually_zero] at C obtain ⟨u₁, h₁u₁, h₂u₁⟩ := h₂f rw [(h₁f.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hu C) h₁u₁] at h₂u₁ tauto theorem discreteZeros {f : ℂ → ℂ} {U : Set ℂ} (hU : IsPreconnected U) (h₁f : AnalyticOn ℂ f U) (h₂f : ∃ u ∈ U, f u ≠ 0) : DiscreteTopology ((U.restrict f)⁻¹' {0}) := by simp_rw [← singletons_open_iff_discrete] simp_rw [Metric.isOpen_singleton_iff] intro z let A := XX hU h₁f h₂f z.1.2 rw [eq_comm] at A rw [AnalyticAt.order_eq_nat_iff] at A obtain ⟨g, h₁g, h₂g, h₃g⟩ := A rw [Metric.eventually_nhds_iff_ball] at h₃g have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g have : {0}ᶜ ∈ nhds (g z) := by exact compl_singleton_mem_nhds_iff.mpr h₂g let F := h₄g.preimage_mem_nhds this rw [Metric.mem_nhds_iff] at F obtain ⟨ε, h₁ε, h₂ε⟩ := F use ε constructor; exact h₁ε intro y hy let G := h₂ε hy simp at G exact G obtain ⟨ε₁, h₁ε₁⟩ := this obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g use min ε₁ ε₂ constructor · have : 0 < min ε₁ ε₂ := by rw [lt_min_iff] exact And.imp_right (fun _ => h₁ε₂) h₁ε₁ exact this intro y intro h₁y have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by simp calc dist y z _ < min ε₁ ε₂ := by assumption _ ≤ ε₂ := by exact min_le_right ε₁ ε₂ have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by simp calc dist y z _ < min ε₁ ε₂ := by assumption _ ≤ ε₁ := by exact min_le_left ε₁ ε₂ have F := h₂ε₂ y.1 h₂y have : f y = 0 := by exact y.2 rw [this] at F simp at F have : g y.1 ≠ 0 := by exact h₁ε₁.2 y h₃y simp [this] at F ext rw [sub_eq_zero] at F tauto theorem finiteZeros {f : ℂ → ℂ} {U : Set ℂ} (h₁U : IsPreconnected U) (h₂U : IsCompact U) (h₁f : AnalyticOn ℂ f U) (h₂f : ∃ u ∈ U, f u ≠ 0) : Set.Finite ((U.restrict f)⁻¹' {0}) := by have hinter : IsCompact ↑(U ∩ f⁻¹' {0}) := by apply IsCompact.of_isClosed_subset h₂U apply h₁f.continuousOn.preimage_isClosed_of_isClosed exact IsCompact.isClosed h₂U exact isClosed_singleton exact Set.inter_subset_left apply hinter.finite apply DiscreteTopology.of_subset (s := ↑(U ∩ f⁻¹' {0})) exact discreteZeros h₁U h₁f h₂f rfl theorem AnalyticOnCompact.eliminateZeros {f : ℂ → ℂ} {U : Set ℂ} (h₁U : IsPreconnected U) (h₂U : IsCompact U) (h₁f : AnalyticOn ℂ f U) (h₂f : ∃ u ∈ U, f u ≠ 0) : ∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f a a.2).order.toNat) • g z := by let ι : U → ℂ := Subtype.val let A₁ := ι⁻¹' (U ∩ f⁻¹' {0}) have : A₁.Finite := by apply Set.Finite.preimage exact Set.injOn_subtype_val exact finiteZeros h₁U h₂U h₁f h₂f let A := this.toFinset let n : ℂ → ℕ := by intro z by_cases hz : z ∈ U · exact (h₁f z hz).order.toNat · exact 0 have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by intro a _ dsimp [n] simp rw [eq_comm] apply XX h₁U exact h₂f obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn use g use A have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by intro z rw [h₃g z] congr funext a congr dsimp [n] simp [a.2] constructor · exact h₁g · constructor · intro z h₁z by_cases h₂z : ⟨z, h₁z⟩ ∈ ↑A.toSet · exact h₂g ⟨z, h₁z⟩ h₂z · have : f z ≠ 0 := by by_contra C have : ⟨z, h₁z⟩ ∈ ↑A₁ := by dsimp [A₁, ι] simp exact C have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by dsimp [A] simp exact this tauto rw [inter z] at this exact right_ne_zero_of_smul this · exact inter