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Stefan Kebekus
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@ -233,3 +233,216 @@ theorem AnalyticOn.eliminateZeros
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rw [Finset.prod_insert]
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rw [Finset.prod_insert]
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ring
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ring
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exact hb
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exact hb
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theorem XX
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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∀ (hu : u ∈ U), (h₁f u hu).order.toNat = (h₁f u hu).order := by
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intro hu
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apply ENat.coe_toNat
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by_contra C
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rw [(h₁f u hu).order_eq_top_iff] at C
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rw [← (h₁f u hu).frequently_zero_iff_eventually_zero] at C
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obtain ⟨u₁, h₁u₁, h₂u₁⟩ := h₂f
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rw [(h₁f.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hu C) h₁u₁] at h₂u₁
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tauto
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theorem discreteZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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DiscreteTopology ↑(U ∩ f⁻¹' {0}) := by
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simp_rw [← singletons_open_iff_discrete]
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simp_rw [Metric.isOpen_singleton_iff]
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intro z
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let A := XX hU h₁f h₂f z.2.1
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rw [eq_comm] at A
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rw [AnalyticAt.order_eq_nat_iff] at A
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := A
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rw [Metric.eventually_nhds_iff_ball] at h₃g
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have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by
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have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g
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have : {0}ᶜ ∈ nhds (g z) := by
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exact compl_singleton_mem_nhds_iff.mpr h₂g
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let F := h₄g.preimage_mem_nhds this
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rw [Metric.mem_nhds_iff] at F
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obtain ⟨ε, h₁ε, h₂ε⟩ := F
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use ε
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constructor; exact h₁ε
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intro y hy
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let G := h₂ε hy
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simp at G
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exact G
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obtain ⟨ε₁, h₁ε₁⟩ := this
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obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
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use min ε₁ ε₂
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constructor
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· have : 0 < min ε₁ ε₂ := by
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rw [lt_min_iff]
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exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
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exact this
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intro y
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intro h₁y
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have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₂ := by exact min_le_right ε₁ ε₂
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have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₁ := by exact min_le_left ε₁ ε₂
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have F := h₂ε₂ y.1 h₂y
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rw [y.2.2] at F
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simp at F
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have : g y.1 ≠ 0 := by
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exact h₁ε₁.2 y h₃y
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simp [this] at F
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ext
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rw [sub_eq_zero] at F
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tauto
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theorem finiteZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsPreconnected U)
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(h₂U : IsCompact U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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Set.Finite ↑(U ∩ f⁻¹' {0}) := by
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have hinter : IsCompact ↑(U ∩ f⁻¹' {0}) := by
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apply IsCompact.of_isClosed_subset h₂U
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apply h₁f.continuousOn.preimage_isClosed_of_isClosed
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exact IsCompact.isClosed h₂U
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exact isClosed_singleton
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exact Set.inter_subset_left
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apply hinter.finite
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apply DiscreteTopology.of_subset (s := ↑(U ∩ f⁻¹' {0}))
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exact discreteZeros h₁U h₁f h₂f
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rfl
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theorem AnalyticOnCompact.eliminateZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsPreconnected U)
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(h₂U : IsCompact U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ u ∈ U, f u ≠ 0) :
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∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ U, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
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let A := U ∩ f ⁻¹' {0}
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by sorry -- (finiteZeros h₁U h₂U h₁f h₂f).toFinset
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let B := AnalyticOn.eliminateZeros h₁f
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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)
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-- case empty
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simp
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use f
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simp
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exact hf
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-- case insert
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intro b₀ B hb iHyp
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intro hBinsert
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obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha))
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have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
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rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)]
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let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a.1)
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have : f = fun z ↦ φ z * g₀ z := by
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funext z
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rw [h₃g₀ z]
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rfl
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simp_rw [this]
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have h₁φ : AnalyticAt ℂ φ b₀ := by
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dsimp [φ]
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apply Finset.analyticAt_prod
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intro b _
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id ℂ
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exact analyticAt_const
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have h₂φ : h₁φ.order = (0 : ℕ) := by
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rw [AnalyticAt.order_eq_nat_iff h₁φ 0]
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use φ
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constructor
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· assumption
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· constructor
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· dsimp [φ]
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push_neg
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rw [Finset.prod_ne_zero_iff]
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intro a ha
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simp
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have : ¬ (b₀.1 - a.1 = 0) := by
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by_contra C
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rw [sub_eq_zero] at C
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rw [SetCoe.ext C] at hb
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tauto
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tauto
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· simp
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rw [AnalyticAt.order_mul h₁φ (h₁g₀ b₀ b₀.2)]
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rw [h₂φ]
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simp
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obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this
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use g₁
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constructor
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· exact h₁g₁
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· constructor
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· intro a h₁a
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by_cases h₂a : a = b₀
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· rwa [h₂a]
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· let A' := Finset.mem_of_mem_insert_of_ne h₁a h₂a
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let B' := h₃g₁ a
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let C' := h₂g₀ a A'
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rw [B'] at C'
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exact right_ne_zero_of_smul C'
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· intro z
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let A' := h₃g₀ z
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rw [h₃g₁ z] at A'
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rw [A']
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rw [← smul_assoc]
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congr
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simp
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rw [Finset.prod_insert]
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ring
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exact hb
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@ -282,186 +282,6 @@ theorem zeroDivisor_finiteOnCompact
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exact Set.inter_subset_right
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exact Set.inter_subset_right
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theorem AnalyticOn.order_eq_nat_iff
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{z₀ : ℂ}
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(hf : AnalyticOn ℂ f U)
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(hz₀ : z₀ ∈ U)
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(n : ℕ) :
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(hf z₀ hz₀).order = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
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constructor
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-- Direction →
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intro hn
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obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
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-- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
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-- removable singularity removed
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let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
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-- Describe g near z₀
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have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
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rw [eventually_nhds_iff]
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
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use t
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constructor
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· intro y h₁y
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by_cases h₂y : y = z₀
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· dsimp [g]; simp [h₂y]
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· dsimp [g]; simp [h₂y]
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rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
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exact h₁t y h₁y
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norm_num
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rw [sub_eq_zero]
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tauto
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· constructor
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· assumption
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· assumption
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-- Describe g near points z₁ that are different from z₀
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have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
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intro hz₁
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rw [eventually_nhds_iff]
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use {z₀}ᶜ
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constructor
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· intro y hy
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simp at hy
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simp [g, hy]
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· exact ⟨isOpen_compl_singleton, hz₁⟩
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-- Use g and show that it has all required properties
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use g
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constructor
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· -- AnalyticOn ℂ g U
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intro z h₁z
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by_cases h₂z : z = z₀
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· rw [h₂z]
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apply AnalyticAt.congr h₁gloc
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exact Filter.EventuallyEq.symm g_near_z₀
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· simp_rw [eq_comm] at g_near_z₁
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apply AnalyticAt.congr _ (g_near_z₁ h₂z)
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apply AnalyticAt.div
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exact hf z h₁z
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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simp
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rw [sub_eq_zero]
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tauto
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· constructor
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· simp [g]; tauto
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· intro z
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by_cases h₂z : z = z₀
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· rw [h₂z, g_near_z₀.self_of_nhds]
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exact h₃gloc.self_of_nhds
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· rw [(g_near_z₁ h₂z).self_of_nhds]
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simp [h₂z]
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rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel]
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simp; norm_num
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rw [sub_eq_zero]
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tauto
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-- direction ←
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intro h
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
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rw [AnalyticAt.order_eq_nat_iff]
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use g
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exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.eventually_of_forall h₃g⟩⟩
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theorem AnalyticOn.order_eq_nat_iff'
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{A : Finset U}
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(hf : AnalyticOn ℂ f U)
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(n : A → ℕ) :
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∀ a : A, (hf a (Subtype.coe_prop a.val)).order = n a → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a, g a ≠ 0) ∧ ∀ z, f z = (∏ a, (z - a) ^ (n a)) • g z := by
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apply Finset.induction
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let a : A := by sorry
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let b : ℂ := by sorry
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let u : U := by sorry
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let X := n a
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have : a = (3 : ℂ) := by sorry
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have : b ∈ ↑A := by sorry
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have : ↑a ∈ U := by exact Subtype.coe_prop a.val
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let Y := ∀ a : A, (hf a (Subtype.coe_prop a.val)).order = n a
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--∀ a : A, (hf (ha a)).order = ↑(n a) →
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intro hn
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obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
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-- Define a candidate function
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let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
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-- Describe g near z₀
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have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
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rw [eventually_nhds_iff]
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
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use t
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constructor
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· intro y h₁y
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by_cases h₂y : y = z₀
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· dsimp [g]; simp [h₂y]
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· dsimp [g]; simp [h₂y]
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rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
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exact h₁t y h₁y
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norm_num
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rw [sub_eq_zero]
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tauto
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· constructor
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· assumption
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· assumption
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|
|
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-- Describe g near points z₁ 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
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|
||||||
rw [eventually_nhds_iff]
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|
||||||
use {z₀}ᶜ
|
|
||||||
constructor
|
|
||||||
· intro y hy
|
|
||||||
simp at hy
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|
||||||
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_cancel]
|
|
||||||
simp; norm_num
|
|
||||||
rw [sub_eq_zero]
|
|
||||||
tauto
|
|
||||||
|
|
||||||
|
|
||||||
noncomputable def zeroDivisorDegree
|
noncomputable def zeroDivisorDegree
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue