| import OAI.AlgebraicGeometry.PlaneCurves.AnalyticOrder |
| import OAI.AlgebraicGeometry.PlaneCurves.DividedFamilies |
| import OAI.AlgebraicGeometry.PlaneCurves.LineRestriction |
|
|
| namespace OAI |
|
|
| /-! |
| # Nested, parametric, and line order along normal graphs |
| -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology BigOperators |
|
|
| /-- The selected central expression is an actual analytic function near any |
| zero-normal point where the inverse chart is analytic. -/ |
| theorem central_expression_analyticAt |
| (I : Finset ℕ) (j : ℕ → ℕ) (R : ℕ → MvPolynomial (Fin 2) ℂ) (u : ℕ) |
| (κ : ComplexPlane → ComplexPlane) (ξ c : ℂ) |
| (hκ : AnalyticAt ℂ κ (ξ, 0)) : |
| AnalyticAt ℂ |
| (fun q : ComplexPlane => ∑ α ∈ I.filter (fun α => α + j α = u), |
| q.2 ^ j α * planePolynomialEval (R α) (κ (q.1, 0))) (ξ, c) := by |
| have hf := (ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt (ξ, c) |
| have hw := (ContinuousLinearMap.snd ℂ ℂ ℂ).analyticAt (ξ, c) |
| have harg : AnalyticAt ℂ (fun q : ComplexPlane => (q.1, (0 : ℂ))) (ξ, c) := |
| hf.prod analyticAt_const |
| have hc := hκ.comp (f := fun q : ComplexPlane => (q.1, (0 : ℂ))) harg |
| apply Finset.analyticAt_fun_sum |
| intro α hα |
| have hr : AnalyticAt ℂ (fun q : ComplexPlane => planePolynomialEval (R α) (κ (q.1, 0))) (ξ, c) := |
| AnalyticAt.comp (g := planePolynomialEval (R α)) |
| (f := fun q : ComplexPlane => κ (q.1, 0)) |
| (planePolynomialEval_analyticAt (R α) (κ (ξ, 0))) hc |
| exact (hw.pow (j α)).mul hr |
|
|
| /-- The actual central derivative conclusion implies actual analytic order |
| for the graph-restricted polynomial expression, with the local graph equality |
| proved from the normal chart rather than assumed as a representation field. -/ |
| theorem graph_expression_order_of_central_jets |
| (I : Finset ℕ) (j : ℕ → ℕ) (R : ℕ → MvPolynomial (Fin 2) ℂ) (u : ℕ) |
| (G : MvPolynomial (Fin 2) ℂ) |
| (e : OpenPartialHomeomorph ComplexPlane ComplexPlane) |
| (he : (e : ComplexPlane → ComplexPlane) = polynomialNormalCoordinates G) |
| (b : ℂ → ℂ) (ξ c : ℂ) (hb : ContinuousAt b ξ) |
| (hsource : (ξ, b ξ) ∈ e.source) |
| (hG : ∀ᶠ z in 𝓝 ξ, planePolynomialEval G (z, b z) = 0) |
| (hi : AnalyticAt ℂ e.symm (ξ, 0)) (m : ℕ) |
| (hjets : ∀ n < m, iteratedFDeriv ℂ n |
| (fun q : ComplexPlane => ∑ α ∈ I.filter (fun α => α + j α = u), |
| q.2 ^ j α * planePolynomialEval (R α) (e.symm (q.1, 0))) (ξ, c) = 0) : |
| HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I.filter (fun α => α + j α = u), |
| q.2 ^ j α * planePolynomialEval (R α) (q.1, b q.1)) (ξ, c) m := by |
| have h := Nagata.Workers.W17.analytic_order_of_iteratedFDeriv_zero |
| (central_expression_analyticAt I j R u e.symm ξ c hi) m hjets |
| exact h.congr (central_expression_eq_graph_locally I j R u G e he b ξ c hb hsource hG) |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology BigOperators |
|
|
| /-- The actual normal polynomial with univariate graph-restricted coefficients. -/ |
| def selectedGraphPolynomial (I : Finset ℕ) (j : ℕ → ℕ) |
| (B : ℕ → Polynomial ℂ) (u : ℕ) : Polynomial (Polynomial ℂ) := |
| ∑ α ∈ I.filter (fun α => α + j α = u), Polynomial.monomial (j α) (B α) |
|
|
| theorem eval_selectedGraphPolynomial |
| (I : Finset ℕ) (j : ℕ → ℕ) (B : ℕ → Polynomial ℂ) (u : ℕ) (q : ComplexPlane) : |
| ((selectedGraphPolynomial I j B u).eval (Polynomial.C q.2)).eval q.1 = |
| ∑ α ∈ I.filter (fun α => α + j α = u), q.2 ^ j α * (B α).eval q.1 := by |
| simp [selectedGraphPolynomial, Polynomial.eval_finsetSum, mul_comm] |
|
|
| /-- The genuine local graph analytic order gives ordinary ideal-power |
| multiplicity for its actual finite normal polynomial, via exact coefficient evaluation. -/ |
| theorem selectedGraphPolynomial_order_of_graph_order |
| (I : Finset ℕ) (j : ℕ → ℕ) (R : ℕ → MvPolynomial (Fin 2) ℂ) |
| (B : ℕ → Polynomial ℂ) (u : ℕ) (b : ℂ → ℂ) (ξ c : ℂ) (m : ℕ) |
| (hR : ∀ α ∈ I.filter (fun α => α + j α = u), ∀ z : ℂ, |
| planePolynomialEval (R α) (z, b z) = (B α).eval z) |
| (horder : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I.filter (fun α => α + j α = u), |
| q.2 ^ j α * planePolynomialEval (R α) (q.1, b q.1)) (ξ, c) m) : |
| selectedGraphPolynomial I j B u ∈ (Nagata.W18.nestedPointIdeal ξ c) ^ m := by |
| apply Nagata.W18.nested_polynomial_ideal_order_of_analytic_order _ ξ c m |
| apply horder.congr |
| exact Filter.Eventually.of_forall fun q => by |
| dsimp only |
| rw [eval_selectedGraphPolynomial] |
| apply Finset.sum_congr rfl |
| intro α hα |
| rw [hR α hα q.1] |
|
|
| /-- Coefficientwise restriction of the actual selected plane normal polynomial |
| is literally the graph normal polynomial used above. -/ |
| theorem map_selected_monomial_sum |
| {A : Type*} [CommRing A] (φ : A →+* Polynomial ℂ) |
| (I : Finset ℕ) (j : ℕ → ℕ) (T : ℕ → A) (u : ℕ) : |
| Polynomial.map φ |
| (∑ α ∈ I.filter (fun α => α + j α = u), Polynomial.monomial (j α) (T α)) = |
| selectedGraphPolynomial I j (fun α => φ (T α)) u := by |
| simp [selectedGraphPolynomial, Polynomial.map_sum] |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology BigOperators |
|
|
| /-- An actual curve parametrization on G=0 agrees locally with the zero-normal |
| slice of the normal chart. No inverse for the tangential parametrization is used. -/ |
| theorem inverse_normal_chart_eq_parametric_curve |
| (G : MvPolynomial (Fin 2) ℂ) |
| (e : OpenPartialHomeomorph ComplexPlane ComplexPlane) |
| (he : (e : ComplexPlane → ComplexPlane) = polynomialNormalCoordinates G) |
| (p : ℂ → ComplexPlane) (z₀ : ℂ) (hp : ContinuousAt p z₀) |
| (hsource : p z₀ ∈ e.source) |
| (hG : ∀ᶠ z in 𝓝 z₀, planePolynomialEval G (p z) = 0) : |
| ∀ᶠ z in 𝓝 z₀, e.symm ((p z).1, 0) = p z := by |
| have hs : ∀ᶠ z in 𝓝 z₀, p z ∈ e.source := |
| hp.preimage_mem_nhds (e.open_source.mem_nhds hsource) |
| filter_upwards [hs, hG] with z hz hg |
| have hh := e.left_inv hz |
| simpa only [he, polynomialNormalCoordinates, hg] using hh |
|
|
| /-- Actual reparametrization and homogeneous normal-fiber rescaling. -/ |
| def curveNormalParameter (p : ℂ → ComplexPlane) (a : ℂ → ℂ) (k : ℕ) |
| (q : ComplexPlane) : ComplexPlane := ((p q.1).1, q.2 / a q.1 ^ k) |
|
|
| theorem curveNormalParameter_analyticAt |
| (p : ℂ → ComplexPlane) (a : ℂ → ℂ) (k : ℕ) (z₀ w₀ : ℂ) |
| (hp : AnalyticAt ℂ p z₀) (ha : AnalyticAt ℂ a z₀) (ha0 : a z₀ ≠ 0) : |
| AnalyticAt ℂ (curveNormalParameter p a k) (z₀, w₀) := by |
| have hf := (ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt (z₀, w₀) |
| have hw := (ContinuousLinearMap.snd ℂ ℂ ℂ).analyticAt (z₀, w₀) |
| have hpc := hp.comp (f := Prod.fst) hf |
| have ht := ((ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt (p z₀)).comp (f := fun q : ComplexPlane => p q.1) hpc |
| have hac := ha.comp (f := Prod.fst) hf |
| exact ht.prod (hw.div (hac.pow k) (pow_ne_zero k ha0)) |
|
|
| /-- Actual central normal order pulls back along an analytic curve and is |
| preserved by multiplication with the actual homogeneous frame factor a(z)^d. -/ |
| theorem parametric_frame_order_of_central_order |
| (I : Finset ℕ) (j : ℕ → ℕ) (R : ℕ → MvPolynomial (Fin 2) ℂ) |
| (G : MvPolynomial (Fin 2) ℂ) |
| (e : OpenPartialHomeomorph ComplexPlane ComplexPlane) |
| (he : (e : ComplexPlane → ComplexPlane) = polynomialNormalCoordinates G) |
| (p : ℂ → ComplexPlane) (a : ℂ → ℂ) (d k : ℕ) (z₀ w₀ : ℂ) |
| (hp : AnalyticAt ℂ p z₀) (ha : AnalyticAt ℂ a z₀) (ha0 : a z₀ ≠ 0) |
| (hsource : p z₀ ∈ e.source) |
| (hG : ∀ᶠ z in 𝓝 z₀, planePolynomialEval G (p z) = 0) |
| (m : ℕ) |
| (horder : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I, |
| q.2 ^ j α * planePolynomialEval (R α) (e.symm (q.1, 0))) |
| ((p z₀).1, w₀ / a z₀ ^ k) m) : |
| HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => a q.1 ^ d * ∑ α ∈ I, |
| (q.2 / a q.1 ^ k) ^ j α * planePolynomialEval (R α) (p q.1)) (z₀, w₀) m := by |
| have hc := horder.comp (f := curveNormalParameter p a k) (curveNormalParameter_analyticAt p a k z₀ w₀ hp ha ha0) |
| have hg := inverse_normal_chart_eq_parametric_curve G e he p z₀ hp.continuousAt hsource hG |
| have hf := (ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt (z₀, w₀) |
| have hge := hf.continuousAt.tendsto.eventually hg |
| have hlocal : HasAnalyticOrderAtLeast (𝕜 := ℂ) |
| (fun q : ComplexPlane => ∑ α ∈ I, |
| (q.2 / a q.1 ^ k) ^ j α * planePolynomialEval (R α) (p q.1)) (z₀, w₀) m := by |
| apply hc.congr |
| exact hge.mono fun q hq => by |
| change e.symm ((p q.1).1, 0) = p q.1 at hq |
| simp only [Function.comp_apply, curveNormalParameter, hq] |
| exact Nagata.VariableUnitOrder.analytic_order_mul_left |
| ((ha.comp (f := Prod.fst) hf).pow d) hlocal |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Nagata.Workers.W28 |
| open scoped Topology BigOperators |
|
|
| /-- The actual selected homogeneous normal polynomial, restricted to a line, |
| has ordinary surface multiplicity forced by the local normal specialization. |
| All coefficient/frame identifications use the actual Z=1 chart homomorphism |
| and actual homogeneous line-restriction homomorphism. -/ |
| theorem line_restricted_initial_order |
| (I : Finset ℕ) (j : ℕ → ℕ) (T : ℕ → MvPolynomial (Fin 3) ℂ) (u : ℕ) |
| (G : MvPolynomial (Fin 3) ℂ) (i ξ c : ℂ) |
| (hline : Nagata.W04.ReducibleSquare.lineRestriction i G = 0) |
| (e : OpenPartialHomeomorph ComplexPlane ComplexPlane) |
| (he : (e : ComplexPlane → ComplexPlane) = |
| polynomialNormalCoordinates (Nagata.W27.directChartHom (2 : Fin 3) G)) |
| (hsource : (ξ, i ^ 2 - i * ξ) ∈ e.source) |
| (hi : AnalyticAt ℂ e.symm (ξ, 0)) (m : ℕ) |
| (hjets : ∀ n < m, iteratedFDeriv ℂ n |
| (fun q : ComplexPlane => ∑ α ∈ I.filter (fun α => α + j α = u), |
| q.2 ^ j α * planePolynomialEval (Nagata.W27.directChartHom (2 : Fin 3) (T α)) |
| (e.symm (q.1, 0))) (ξ, c) = 0) : |
| Polynomial.map (Nagata.W04.ReducibleSquare.lineRestriction i) |
| (∑ α ∈ I.filter (fun α => α + j α = u), Polynomial.monomial (j α) (T α)) ∈ |
| (Nagata.W18.nestedPointIdeal ξ c) ^ m := by |
| let R : ℕ → MvPolynomial (Fin 2) ℂ := fun α => Nagata.W27.directChartHom (2 : Fin 3) (T α) |
| let b : ℂ → ℂ := fun z => i ^ 2 - i * z |
| have hb : ContinuousAt b ξ := |
| (continuous_const.sub (continuous_const.mul continuous_id)).continuousAt |
| have hg : ∀ᶠ z in 𝓝 ξ, |
| planePolynomialEval (Nagata.W27.directChartHom (2 : Fin 3) G) (z, b z) = 0 := by |
| exact Filter.Eventually.of_forall fun z => by |
| change planePolynomialEval (Nagata.W27.directChartHom (2 : Fin 3) G) (z, i ^ 2 - i * z) = 0 |
| rw [Nagata.Workers.W24.planePolynomialEval_directChart_on_line, hline] |
| simp |
| have horder := graph_expression_order_of_central_jets I j R u |
| (Nagata.W27.directChartHom (2 : Fin 3) G) e he b ξ c hb hsource hg hi m hjets |
| rw [map_selected_monomial_sum] |
| apply selectedGraphPolynomial_order_of_graph_order I j R |
| (fun α => Nagata.W04.ReducibleSquare.lineRestriction i (T α)) u b ξ c m _ horder |
| intro α hα z |
| exact Nagata.Workers.W24.planePolynomialEval_directChart_on_line (T α) i z |
|
|
| end Nagata.Workers.W28 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
