| import OAI.NumberTheory.PowerFree.RadialFibres |
|
|
| namespace OAI |
|
|
| namespace QuarticPowerFree |
|
|
| section |
| open Filter NumberField |
| open scoped Topology Classical BigOperators |
|
|
| theorem indexed_card_le_sum_fibres {ι κ : Type*} [Fintype ι] [Fintype κ] |
| (b : ι → κ) : Fintype.card ι ≤ ∑ j : κ, (Finset.univ.filter (fun i => b i=j)).card := by |
| classical |
| let R (j : κ) := Finset.univ.filter (fun i => b i=j) |
| have hcover : (Finset.univ : Finset ι) ⊆ Finset.univ.biUnion R := by |
| intro i _ |
| exact Finset.mem_biUnion.mpr ⟨b i,Finset.mem_univ _,Finset.mem_filter.mpr ⟨Finset.mem_univ _,rfl⟩⟩ |
| simpa only [Finset.card_univ] using (Finset.card_le_card hcover).trans Finset.card_biUnion_le |
|
|
| theorem quartic_dyadic_tail (f : Polynomial ℤ) (hf : f.IsPrimitive) |
| (hirr : Irreducible (f.map (Int.castRingHom ℚ))) (hc : 0 < f.leadingCoeff) |
| (hd : f.natDegree=4) : |
| ∃ C : ℝ, 0 < C ∧ ∀ᶠ N : ℕ in atTop, |
| (dyadicTailCount f (f.natDegree-2) N:ℝ) ≤ C*(N:ℝ)^(1-3/1000:ℝ) := by |
| classical |
| obtain ⟨M,μ,Q,C,Xcut,hμ,hC,_hXcut,hQ,hfactor⟩ := |
| balanced_arithmetic f hirr (by omega) hc (f.natDegree-2) (by omega) |
| obtain ⟨F,hF,heval⟩ := polynomial_dyadic_eval_bound f |
| have hK : Module.finrank ℚ M.Carrier=4 := M.degree_eq.trans hd |
| let T := integralEmbeddingChart M.Carrier 4 hK |
| have hθ : Function.Injective (fun i => T.σ i M.root) := by |
| intro i j hij |
| apply T.σ.injective |
| apply (RingHom.equivRatAlgHom M.Carrier ℂ).injective |
| apply M.powerBasis.algHom_ext |
| change T.σ i M.powerBasis.gen = T.σ j M.powerBasis.gen |
| rw [M.powerBasis_gen,M.integralRoot_eq] |
| simp only [map_mul,map_intCast,hij] |
| have hboxes (j : Fin 100) := balanced_coordinates_quarticbox j T μ hμ C F hC (by linarith) |
| choose Cbox hCbox hbox using hboxes |
| have hcounts (j : Fin 100) := quartic_indexed_count f hf hd j hK T M.root hθ μ hμ Q hQ (Cbox j) (hCbox j) |
| choose D Xcount hD hcount using hcounts |
| let Dall : ℝ := ∑ j : Fin 100, D j |
| have hDall : 0 < Dall := Finset.sum_pos (fun j _ => hD j) Finset.univ_nonempty |
| refine ⟨Dall,hDall,?_⟩ |
| have hlim : Tendsto (fun N : ℕ => (N:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop |
| have hall : ∀ᶠ N : ℕ in atTop, ∀ j : Fin 100, Xcount j < (N:ℝ) := |
| Filter.eventually_all.mpr (fun j => hlim.eventually (eventually_gt_atTop (Xcount j))) |
| filter_upwards [hlim.eventually (eventually_gt_atTop (max 1 Xcut)),hall] with N hN hNcount |
| have hN1 : 1 ≤ (N:ℝ) := (le_max_left _ _).trans hN.le |
| have hNcut : Xcut ≤ (N:ℝ) := (le_max_right _ _).trans hN.le |
| let S := (Finset.Ioc N (2*N)).filter fun n : ℕ => |
| ∃ p : ℕ, p.Prime ∧ N < p ∧ (p:ℤ)^(f.natDegree-2) ∣ f.eval (n:ℤ) |
| have hS (n : S) : ∃ p : ℕ, p.Prime ∧ N < p ∧ |
| (p:ℤ)^(f.natDegree-2) ∣ f.eval (n.val:ℤ) := (Finset.mem_filter.mp n.property).2 |
| choose P hP hNP hdiv using hS |
| have hrange (n : S) : (N:ℝ) < (n.val:ℝ) ∧ (n.val:ℝ) ≤ 2*(N:ℝ) := by |
| have hn := Finset.mem_Ioc.mp (Finset.mem_filter.mp n.property).1 |
| constructor |
| · exact_mod_cast hn.1 |
| · exact_mod_cast hn.2 |
| have hNP' (n : S) : (N:ℝ) < P n := by exact_mod_cast hNP n |
| have hfac (n : S) := (hfactor N hNcut (n.val:ℤ) (P n) |
| (by exact_mod_cast (hrange n).1) (by exact_mod_cast (hrange n).2) |
| (hP n) (hNP' n) (hdiv n)).2 |
| choose J hJ A hA B hAB hnorm harch using hfac |
| have hprime (n : S) : (P n).Prime ∧ (N:ℝ) < P n ∧ (P n:ℤ) ∣ f.eval (n.val:ℤ) := by |
| refine ⟨hP n,hNP' n,?_⟩ |
| exact dvd_trans (dvd_pow_self _ (by omega : f.natDegree-2 ≠ 0)) (hdiv n) |
| have hupper (n : S) : (P n:ℝ) ≤ F*(N:ℝ)^2 := by |
| have hp2 := large_prime_power_upper f hirr (by omega) F N (n.val:ℤ) (P n) (f.natDegree-2) |
| (heval N hN1 (n.val:ℤ) (by positivity) (by exact_mod_cast (hrange n).2)) (hdiv n) |
| rw [hd] at hp2 |
| simp only [Nat.reduceSub,Real.rpow_natCast] at hp2 |
| change (P n:ℝ)^2 ≤ F*(N:ℝ)^(4:ℕ) at hp2 |
| apply (sq_le_sq₀ (by positivity : (0:ℝ) ≤ P n) (by positivity : 0 ≤ F*(N:ℝ)^2)).mp |
| have hFF : F ≤ F^2 := by nlinarith |
| have hh := mul_le_mul_of_nonneg_right hFF (sq_nonneg ((N:ℝ)^2)) |
| nlinarith only [hp2,hh] |
| have hbins (n : S) := quartic_prime_bin N F (P n) hN1 (by linarith) (hNP' n) (hupper n) |
| choose bin hbinlo hbinhi using hbins |
| let R (j : Fin 100) := Finset.univ.filter (fun n : S => bin n=j) |
| have hR (j : Fin 100) : ((R j).card:ℝ) ≤ D j*(N:ℝ)^(997/1000:ℝ) := by |
| let a : R j → ℤ := fun n => (n.val.val:ℤ) |
| have hainj : Function.Injective a := by |
| intro n m he |
| apply Subtype.ext |
| apply Subtype.ext |
| exact Int.natCast_inj.mp he |
| have hbe (n : R j) : bin n.val=j := (Finset.mem_filter.mp n.property).2 |
| have hcoords (n : R j) (i) : |(T.coord (A n.val) i:ℝ)| ≤ Cbox j*(N:ℝ)^(quarticU j) ∧ |
| |(T.coord (B n.val) i:ℝ)| ≤ Cbox j*(N:ℝ)^(quarticV j) := by |
| have hlo : (N:ℝ)^(quarticLowerEta j) < P n.val := by simpa only [hbe n] using hbinlo n.val |
| have hhi : (P n.val:ℝ) ≤ F*(N:ℝ)^(quarticUpperEta j) := by simpa only [hbe n] using hbinhi n.val |
| apply hbox j N hN1 (P n.val) hlo hhi (A n.val) (B n.val) ?_ i |
| intro w |
| have hh := And.intro (harch n.val w).1.2 (harch n.val w).2.2 |
| simpa only [hd,Nat.reduceSub,Nat.cast_ofNat] using hh |
| have hh := hcount j N (hNcount j) (R j) a (fun n => P n.val) (fun n => J n.val) |
| (fun n => A n.val) (fun n => B n.val) hainj |
| (fun n => by simpa only [a,Int.cast_natCast] using hrange n.val) |
| (fun n => hJ n.val) (fun n => hprime n.val) (fun n => hA n.val) |
| (fun n => by simpa only [hd,Nat.reduceSub] using hAB n.val) |
| (fun n => hnorm n.val) hcoords |
| simpa only [Fintype.card_coe] using hh |
| have hcard : (S.card:ℝ) ≤ ∑ j : Fin 100, ((R j).card:ℝ) := by |
| have hh := (Nat.cast_le (α := ℝ)).mpr (indexed_card_le_sum_fibres bin) |
| simp only [Fintype.card_coe,Nat.cast_sum] at hh |
| convert hh using 1 |
| apply Finset.sum_congr rfl |
| intro j _ |
| congr 2 |
| ext n |
| simp only [R,Finset.mem_filter,Finset.mem_univ,true_and] |
| calc |
| (dyadicTailCount f (f.natDegree-2) N:ℝ) = (S.card:ℝ) := rfl |
| _ ≤ ∑ j : Fin 100, ((R j).card:ℝ) := hcard |
| _ ≤ ∑ j : Fin 100, D j*(N:ℝ)^(997/1000:ℝ) := Finset.sum_le_sum (fun j _ => hR j) |
| _ = Dall*(N:ℝ)^(1-3/1000:ℝ) := by rw [←Finset.sum_mul]; norm_num only [show (1-3/1000:ℝ)=997/1000 by norm_num]; rfl |
|
|
| end |
|
|
| theorem density_ge_four (f : Polynomial ℤ) |
| (hirr : Irreducible (f.map (Int.castRingHom ℚ))) (hd : 4 ≤ f.natDegree) |
| (hlocal : LocallyAdmissible f (f.natDegree-2)) : |
| DensityStatement f (f.natDegree-2) := by |
| by_cases hd5 : 5 ≤ f.natDegree |
| · exact density_ge_five f hirr hd5 hlocal |
| obtain ⟨a,g,ha,hfg,hgprim,hgpos,hgdeg,hgirr⟩ := exists_primitive_positive_normalization f hirr |
| obtain ⟨C,_hC,hbound⟩ := quartic_dyadic_tail g hgprim hgirr hgpos (by omega) |
| have ht := dyadic_tail_limit_of_power_saving g (g.natDegree-2) (3/1000) C (by norm_num) hbound |
| rw [hgdeg] at ht |
| exact densityStatement_from_normalized_dyadic_tail f g hirr (by omega) |
| (by omega) hlocal a ha hfg ht |
|
|
| theorem density (f : Polynomial ℤ) |
| (hirr : Irreducible (f.map (Int.castRingHom ℚ))) |
| (hdeglo : 4 ≤ f.natDegree) (hdeghi : f.natDegree ≤ 8) |
| (hlocal : LocallyAdmissible f (f.natDegree - 2)) : |
| DensityStatement f (f.natDegree - 2) := by |
| have _ := hdeghi |
| exact density_ge_four f hirr hdeglo hlocal |
|
|
| theorem allDegrees (f : Polynomial ℤ) |
| (hirr : Irreducible (f.map (Int.castRingHom ℚ))) |
| (hdeglo : 4 ≤ f.natDegree) |
| (hlocal : LocallyAdmissible f (f.natDegree - 2)) : |
| DensityStatement f (f.natDegree - 2) := by |
| exact density_ge_four f hirr hdeglo hlocal |
|
|
| end QuarticPowerFree |
|
|
| end OAI |
|
|
