37 lines
875 B
Plaintext
37 lines
875 B
Plaintext
import Mathlib.Analysis.Analytic.IsolatedZeros
|
||
import Nevanlinna.holomorphic
|
||
|
||
|
||
noncomputable def zeroDivisor
|
||
(f : ℂ → ℂ) :
|
||
ℂ → ℕ := by
|
||
intro z
|
||
if hf : AnalyticAt ℂ f z then
|
||
exact hf.order.toNat
|
||
else
|
||
exact 0
|
||
|
||
|
||
theorem discreteZeros
|
||
{f : ℂ → ℂ} :
|
||
DiscreteTopology (Function.support (zeroDivisor f)) := by
|
||
sorry
|
||
|
||
|
||
theorem zeroDivisor_finiteOnCompact
|
||
{f : ℂ → ℂ}
|
||
{s : Set ℂ}
|
||
(hs : IsCompact s) :
|
||
Set.Finite (s ∩ Function.support (zeroDivisor f)) := by
|
||
sorry
|
||
|
||
|
||
theorem eliminatingZeros
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
{R : ℝ}
|
||
(h₁f : ∀ z ∈ Metric.ball z₀ R, HolomorphicAt f z)
|
||
(h₂f : ∃ z ∈ Metric.ball z₀ R, f z ≠ 0) :
|
||
∃ F : ℂ → ℂ, ∀ z ∈ Metric.ball z₀ R, (HolomorphicAt F z) ∧ (f z = (F z) * ∏ᶠ a ∈ Metric.ball z₀ R, (z - a) ^ (zeroDivisor f a) ) := by
|
||
sorry
|