| import OAI.NumberTheory.PiExponent.Geometry.CurveParameterFinite |
| import OAI.NumberTheory.PiExponent.Geometry.CurveProductFormula |
| import OAI.NumberTheory.PiExponent.Geometry.CurveValuationCenter |
| import OAI.NumberTheory.PiExponent.Jets.DVRBranchJet |
| import OAI.NumberTheory.PiExponent.Polynomials.WeightedPolynomialPole |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace PiExponent.CurveContactSum |
|
|
| open WeightedPolynomialPole WeightedCurveDegree CurveValuationCenter PolynomialPoleBound |
|
|
| theorem contact_sum_le_of_principal_sum |
| {P : Type*} (z : P →₀ ℤ) (pole : P →₀ ℚ) |
| (hz : z.sum (fun _ n => n) = 0) (hpole_nonneg : ∀ p, 0 ≤ pole p) |
| (S : Finset P) (μ : P → ℝ) (N B : ℝ) (hN : 0 ≤ N) |
| (hlocal : ∀ p ∈ S, B * μ p ≤ (z p : ℝ)) |
| (hglobal : ∀ p, -N * (pole p : ℝ) ≤ (z p : ℝ)) : |
| B * (∑ p ∈ S, μ p) ≤ N * pole.sum (fun _ n => (n : ℝ)) := by |
| classical |
| let T := S ∪ z.support ∪ pole.support |
| have hST : S ⊆ T := by intro p hp; simp [T, hp] |
| have hzT : z.support ⊆ T := by intro p hp; simp [T, hp] |
| have hpT : pole.support ⊆ T := by intro p hp; simp [T, hp] |
| have hzsum : (∑ p ∈ T, (z p : ℝ)) = 0 := by |
| have hh := z.sum_of_support_subset hzT (fun _ n => n) (by simp) |
| have hi : (∑ p ∈ T, z p) = 0 := hh.symm.trans hz |
| exact_mod_cast hi |
| have hpsum : (∑ p ∈ T, (pole p : ℝ)) = pole.sum (fun _ n => (n : ℝ)) := |
| (pole.sum_of_support_subset hpT (fun _ n => (n : ℝ)) (by simp)).symm |
| calc |
| B * (∑ p ∈ S, μ p) = ∑ p ∈ S, B * μ p := Finset.mul_sum _ _ _ |
| _ ≤ ∑ p ∈ S, ((z p : ℝ) + N * (pole p : ℝ)) := by |
| apply Finset.sum_le_sum |
| intro p hp |
| exact (hlocal p hp).trans (le_add_of_nonneg_right |
| (mul_nonneg hN (by exact_mod_cast hpole_nonneg p))) |
| _ ≤ ∑ p ∈ T, ((z p : ℝ) + N * (pole p : ℝ)) := by |
| apply Finset.sum_le_sum_of_subset_of_nonneg hST |
| intro p hp hnot |
| have hh := hglobal p |
| linarith |
| _ = N * pole.sum (fun _ n => (n : ℝ)) := by |
| rw [Finset.sum_add_distrib, ← Finset.mul_sum, hzsum, zero_add, hpsum] |
|
|
| theorem contact_sum_le_div_of_principal_sum |
| {P : Type*} (z : P →₀ ℤ) (pole : P →₀ ℚ) |
| (hz : z.sum (fun _ n => n) = 0) (hpole_nonneg : ∀ p, 0 ≤ pole p) |
| (S : Finset P) (μ : P → ℝ) (N B : ℝ) (hN : 0 ≤ N) (hB : 0 < B) |
| (hlocal : ∀ p ∈ S, B * μ p ≤ (z p : ℝ)) |
| (hglobal : ∀ p, -N * (pole p : ℝ) ≤ (z p : ℝ)) : |
| (∑ p ∈ S, μ p) ≤ N / B * pole.sum (fun _ n => (n : ℝ)) := by |
| rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ hB).mpr |
| simpa only [mul_comm B] using |
| contact_sum_le_of_principal_sum z pole hz hpole_nonneg S μ N B hN hlocal hglobal |
|
|
| theorem coordinatePole_eq_zero {E ι : Type*} [Field E] [Fintype ι] |
| (v : AddValuation E (WithTop ℤ)) (x : ι → E) (w : ι → ℚ) |
| (hx : ∀ i, coordinateOrder v (x i) = 0) : coordinatePole v x w = 0 := by |
| unfold coordinatePole weightedPole |
| simp only [hx, Int.cast_zero, neg_zero, zero_div] |
| have he : (fun i : Option ι => i.elim 0 (fun _ => (0 : ℚ))) = fun _ => 0 := by |
| funext i |
| cases i <;> rfl |
| rw [he] |
| exact Finset.sup'_const _ _ |
| |
| theorem coordinatePole_finite_support {E ι P : Type*} [Field E] [Fintype ι] |
| (v : P → AddValuation E (WithTop ℤ)) (x : ι → E) (w : ι → ℚ) |
| (hfinite : ∀ i, Set.Finite {p | coordinateOrder (v p) (x i) ≠ 0}) : |
| Set.Finite {p | coordinatePole (v p) x w ≠ 0} := by |
| have hu : Set.Finite (⋃ i, {p | coordinateOrder (v p) (x i) ≠ 0}) := |
| Set.finite_iUnion (fun i => hfinite i) |
| apply hu.subset |
| intro p hp |
| by_contra hn |
| apply hp |
| apply coordinatePole_eq_zero |
| intro i |
| by_contra hi |
| exact hn (Set.mem_iUnion.mpr ⟨i,hi⟩) |
| |
| theorem exists_constant_of_isAlgebraic {E : Type*} [Field E] [Algebra ℂ E] |
| (f : E) (hf : IsAlgebraic ℂ f) : ∃ c : ℂ, algebraMap ℂ E c = f := by |
| have hm : f ∈ algebraicClosure ℂ E := mem_algebraicClosure_iff.mpr hf |
| rw [IntermediateField.eq_bot_of_isAlgClosed_of_isAlgebraic (algebraicClosure ℂ E)] at hm |
| exact hm |
| |
| theorem frameWord_field_order_lower |
| {A E : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] |
| [Algebra ℂ A] [Algebra.IsIntegral ℂ (IsLocalRing.ResidueField A)] |
| [Field E] [Algebra A E] [IsFractionRing A E] [Algebra ℂ E] [IsScalarTower ℂ A E] |
| {m : ℕ} (v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i) (H : ℚ) |
| (c : Fin m → ℂ) (y : A) (x : Fin m → A) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i) |
| (hnc : y ≠ 1 ∨ ∃ i, x i ≠ algebraMap ℂ A (c i)) |
| (p : PiExponentApprox.FramePolynomial m) |
| (hp : FormalLogJet.formalJet c p ∈ |
| JetGeometry.rationalWeightedIdeal v (fun i => (hv i).le) H) |
| (word : List (Fin (m+1))) |
| (hne : MvPolynomial.aeval (Fin.cases (algebraMap A E y) (fun i => algebraMap A E (x i))) |
| (PiExponentApprox.polynomialFrameWord m word p) ≠ 0) : |
| DVRBranchJet.contact v c y x hnc * (H - (word.map v).sum) ≤ |
| (integerOrder (CurveLocalOrder.fractionAddValuation A E) |
| (Units.mk0 (MvPolynomial.aeval |
| (Fin.cases (algebraMap A E y) (fun i => algebraMap A E (x i))) |
| (PiExponentApprox.polynomialFrameWord m word p)) hne) : ℚ) := by |
| have he (q : PiExponentApprox.FramePolynomial m) : |
| algebraMap A E (MvPolynomial.aeval (Fin.cases y x) q) = |
| MvPolynomial.aeval (Fin.cases (algebraMap A E y) (fun i => algebraMap A E (x i))) q := by |
| have hh := MvPolynomial.comp_aeval_apply (f := Fin.cases y x) |
| (IsScalarTower.toAlgHom ℂ A E) q |
| exact hh.trans (congrArg (fun z => MvPolynomial.aeval z q) (by |
| funext i |
| cases i using Fin.cases <;> rfl)) |
| have hn : MvPolynomial.aeval (Fin.cases y x) |
| (PiExponentApprox.polynomialFrameWord m word p) ≠ 0 := by |
| intro hz |
| apply hne |
| rw [← he, hz, map_zero] |
| have hlen := CurveLocalOrder.integerOrder_field_image_eq_length (K := E) hn |
| have hu : Units.mk0 (MvPolynomial.aeval |
| (Fin.cases (algebraMap A E y) (fun i => algebraMap A E (x i))) |
| (PiExponentApprox.polynomialFrameWord m word p)) hne = |
| Units.mk0 (algebraMap A E (MvPolynomial.aeval (Fin.cases y x) |
| (PiExponentApprox.polynomialFrameWord m word p))) |
| ((map_ne_zero_iff _ (IsFractionRing.injective A E)).mpr hn) := by |
| apply Units.ext |
| exact (he _).symm |
| rw [hu, hlen] |
| simpa only [Int.cast_natCast] using |
| DVRBranchJet.frameWord_colength_lower v hv H c y x hy hx hnc p hp word hn |
| |
| theorem coordinateOrder_constant_eq_zero {E : Type*} [Field E] [Algebra ℂ E] |
| (p : NormalizedPlace ℂ E) (c : ℂ) : coordinateOrder p.valuation (algebraMap ℂ E c) = 0 := by |
| by_cases hc : c = 0 |
| · simp [hc, coordinateOrder] |
| have hm : algebraMap ℂ E c ≠ 0 := (map_ne_zero_iff _ (algebraMap ℂ E).injective).mpr hc |
| rw [coordinateOrder, dite_eq_right hm] |
| exact CurveProductFormula.placeOrder_constant_eq_zero p c hc |
| |
| def coordinateOrderDivisor {E : Type*} [Field E] [Algebra ℂ E] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (f : E) : NormalizedPlace ℂ E →₀ ℤ := by |
| classical |
| exact if ht : Transcendental ℂ f then |
| letI := hfinite f ht |
| CurveProductFormula.principalDivisor f ht |
| else 0 |
| |
| @[simp] theorem coordinateOrderDivisor_apply {E : Type*} [Field E] [Algebra ℂ E] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (f : E) (p : NormalizedPlace ℂ E) : |
| coordinateOrderDivisor hfinite f p = coordinateOrder p.valuation f := by |
| classical |
| by_cases ht : Transcendental ℂ f |
| · let := hfinite f ht |
| simp only [coordinateOrderDivisor, dite_eq_left ht, CurveProductFormula.principalDivisor_apply, |
| coordinateOrder, dite_eq_right ht.ne_zero] |
| rfl |
| · rw [coordinateOrderDivisor, dite_eq_right ht] |
| obtain ⟨c,rfl⟩ := exists_constant_of_isAlgebraic f (not_not.mp ht) |
| exact (coordinateOrder_constant_eq_zero p c).symm |
| |
| theorem coordinateOrderDivisor_degree_zero {E : Type*} [Field E] [Algebra ℂ E] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (f : E) : (coordinateOrderDivisor hfinite f).sum (fun _ n => n) = 0 := by |
| classical |
| by_cases ht : Transcendental ℂ f |
| · let := hfinite f ht |
| rw [coordinateOrderDivisor, dite_eq_left ht] |
| exact CurveProductFormula.principalDivisor_degree_zero f ht |
| · simp [coordinateOrderDivisor, ht] |
| |
| def weightedPoleDivisor {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) : NormalizedPlace ℂ E →₀ ℚ := |
| Finsupp.ofSupportFinite (fun p => coordinatePole p.valuation x w) |
| (coordinatePole_finite_support (fun p : NormalizedPlace ℂ E => p.valuation) x w (by |
| intro i |
| simpa only [Function.HasFiniteSupport, Function.support, coordinateOrderDivisor_apply] using |
| (coordinateOrderDivisor hfinite (x i)).hasFiniteSupport)) |
| |
| @[simp] theorem weightedPoleDivisor_apply {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) (p : NormalizedPlace ℂ E) : |
| weightedPoleDivisor hfinite x w p = coordinatePole p.valuation x w := rfl |
| |
| def weightedDegree {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) : ℝ := |
| (weightedPoleDivisor hfinite x w).sum (fun _ n => (n : ℝ)) |
| |
| theorem weightedDegree_nonneg {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) : 0 ≤ weightedDegree hfinite x w := by |
| apply Finset.sum_nonneg |
| intro p hp |
| change (0 : ℝ) ≤ (coordinatePole p.valuation x w : ℚ) |
| exact_mod_cast weightedPole_nonneg w (fun i => (coordinateOrder p.valuation (x i) : ℚ)) |
| |
| theorem constants_real_nonneg {E : Type*} [Field E] [Algebra ℂ E] |
| (p : NormalizedPlace ℂ E) (c : ℂ) : |
| 0 ≤ realValuation p.valuation (algebraMap ℂ E c) := by |
| have h := (show Monotone (WithTop.map (fun z : ℤ => (z : ℝ))) from |
| (Int.cast_mono (R := ℝ)).withTop_map) (p.constants_nonneg c) |
| have hz : WithTop.map (fun z : ℤ => (z : ℝ)) (0 : WithTop ℤ) = (0 : WithTop ℝ) := by |
| change (((0 : ℤ) : ℝ) : WithTop ℝ) = 0 |
| norm_num |
| rw [hz] at h |
| exact h |
| |
| theorem polynomial_order_lower {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (x : ι → E) (w : ι → ℚ) (hw : ∀ i, 0 < w i) (N : ℝ) |
| (f : MvPolynomial ι ℂ) |
| (hdegree : ∀ d ∈ f.support, Finsupp.weight (fun i => (w i : ℝ)) d ≤ N) |
| (hne : MvPolynomial.aeval x f ≠ 0) (p : NormalizedPlace ℂ E) : |
| -N * (coordinatePole p.valuation x w : ℝ) ≤ |
| (coordinateOrder p.valuation (MvPolynomial.aeval x f) : ℝ) := by |
| have h := PolynomialPoleBound.polynomial_order_lower (realValuation p.valuation) x |
| (fun i => (w i : ℝ)) (coordinatePole p.valuation x w : ℝ) N |
| (by exact_mod_cast weightedPole_nonneg w (fun i => (coordinateOrder p.valuation (x i) : ℚ))) |
| (coordinate_order_lower p.valuation x w hw) (constants_real_nonneg p) f hdegree |
| rw [realValuation_of_ne_zero _ _ hne] at h |
| exact WithTop.coe_le_coe.mp h |
| |
| theorem polynomial_contact_sum_le {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) (hw : ∀ i, 0 < w i) (N B : ℝ) (hN : 0 ≤ N) |
| (f : MvPolynomial ι ℂ) |
| (hdegree : ∀ d ∈ f.support, Finsupp.weight (fun i => (w i : ℝ)) d ≤ N) |
| (hne : MvPolynomial.aeval x f ≠ 0) |
| (S : Finset (NormalizedPlace ℂ E)) (μ : NormalizedPlace ℂ E → ℝ) |
| (hlocal : ∀ p ∈ S, B * μ p ≤ |
| (coordinateOrder p.valuation (MvPolynomial.aeval x f) : ℝ)) : |
| B * (∑ p ∈ S, μ p) ≤ N * weightedDegree hfinite x w := by |
| refine contact_sum_le_of_principal_sum |
| (coordinateOrderDivisor hfinite (MvPolynomial.aeval x f)) (weightedPoleDivisor hfinite x w) |
| (coordinateOrderDivisor_degree_zero hfinite _) ?_ S μ N B hN ?_ ?_ |
| · intro p |
| exact weightedPole_nonneg w (fun i => (coordinateOrder p.valuation (x i) : ℚ)) |
| · intro p hp |
| simpa only [coordinateOrderDivisor_apply] using hlocal p hp |
| · intro p |
| simpa only [weightedPoleDivisor_apply, coordinateOrderDivisor_apply] using |
| polynomial_order_lower x w hw N f hdegree hne p |
| |
| theorem polynomial_contact_sum_le_div {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) (hw : ∀ i, 0 < w i) (N B : ℝ) (hN : 0 ≤ N) (hB : 0 < B) |
| (f : MvPolynomial ι ℂ) |
| (hdegree : ∀ d ∈ f.support, Finsupp.weight (fun i => (w i : ℝ)) d ≤ N) |
| (hne : MvPolynomial.aeval x f ≠ 0) |
| (S : Finset (NormalizedPlace ℂ E)) (μ : NormalizedPlace ℂ E → ℝ) |
| (hlocal : ∀ p ∈ S, B * μ p ≤ |
| (coordinateOrder p.valuation (MvPolynomial.aeval x f) : ℝ)) : |
| (∑ p ∈ S, μ p) ≤ N / B * weightedDegree hfinite x w := by |
| rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ hB).mpr |
| simpa only [mul_comm B] using |
| polynomial_contact_sum_le hfinite x w hw N B hN f hdegree hne S μ hlocal |
| |
| theorem polynomial_eq_zero_of_excess_contact |
| {E ι : Type*} [Field E] [Algebra ℂ E] [Fintype ι] |
| (hfinite : ∀ f : E, Transcendental ℂ f → FiniteDimensional (IntermediateField.adjoin ℂ {f}) E) |
| (x : ι → E) (w : ι → ℚ) (hw : ∀ i, 0 < w i) (N B : ℝ) (hN : 0 ≤ N) |
| (f : MvPolynomial ι ℂ) |
| (hdegree : ∀ d ∈ f.support, Finsupp.weight (fun i => (w i : ℝ)) d ≤ N) |
| (S : Finset (NormalizedPlace ℂ E)) (μ : NormalizedPlace ℂ E → ℝ) |
| (hlocal : MvPolynomial.aeval x f ≠ 0 → ∀ p ∈ S, B * μ p ≤ |
| (coordinateOrder p.valuation (MvPolynomial.aeval x f) : ℝ)) |
| (hexcess : N * weightedDegree hfinite x w < B * (∑ p ∈ S, μ p)) : |
| MvPolynomial.aeval x f = 0 := by |
| by_contra hne |
| exact (not_le_of_gt hexcess) |
| (polynomial_contact_sum_le hfinite x w hw N B hN f hdegree hne S μ (hlocal hne)) |
| |
| end PiExponent.CurveContactSum |
| |
| end |
| |
| end OAI |
| |
