| import OAI.NumberTheory.PowerFree.CurveThresholds |
|
|
| namespace OAI |
|
|
| namespace QuarticPowerFree |
|
|
| section |
| open MvPolynomial |
| open scoped BigOperators Classical |
|
|
| theorem quotient_pair_generates {n d : ℕ} (a b : Fin d → MvPolynomial (Fin n) ℂ) |
| (hg : Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤) |
| (I : Ideal (MvPolynomial (Fin n) ℂ)) : |
| Algebra.adjoin ℂ (Set.range (fun i => Ideal.Quotient.mk I (a i)) ∪ |
| Set.range (fun i => Ideal.Quotient.mk I (b i))) = ⊤ := by |
| let e := Ideal.Quotient.mkₐ ℂ I |
| have hr : Set.range (fun i => Ideal.Quotient.mk I (a i)) ∪ Set.range (fun i => Ideal.Quotient.mk I (b i)) = |
| e '' (Set.range a ∪ Set.range b) := by |
| simp only [Set.image_union,← Set.range_comp] |
| rfl |
| rw [hr,Algebra.adjoin_image,hg] |
| apply top_unique |
| intro z _ |
| obtain ⟨p,rfl⟩ := Ideal.Quotient.mkₐ_surjective ℂ I z |
| exact ⟨p,by trivial,_root_.rfl⟩ |
|
|
| theorem norm_dimension_le {n d : ℕ} (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime] |
| (k : ℕ) (a b : Fin d → MvPolynomial (Fin n) ℂ) (q : MvPolynomial (Fin n) ℂ) (θ : Fin d → ℂ) |
| (ha : ∀ i, Ideal.Quotient.mk I (a i) ≠ 0) |
| (heq : ∀ i, Ideal.Quotient.mk I (a i)^k*Ideal.Quotient.mk I (b i) = |
| Ideal.Quotient.mk I q-algebraMap ℂ _ (θ i)) |
| (hg : Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤) |
| (hq : Transcendental ℂ (Ideal.Quotient.mk I q)) : affineDimension I ≤ d+1 := by |
| let := pair_projection_isAlgebraic k (fun i => Ideal.Quotient.mk I (a i)) |
| (fun i => Ideal.Quotient.mk I (b i)) (Ideal.Quotient.mk I q) θ ha heq (quotient_pair_generates a b hg I) |
| obtain ⟨m,j,_,hj,ht⟩ := left_coordinates_transcendence_selection |
| (fun i => Ideal.Quotient.mk I (a i)) (Ideal.Quotient.mk I q) hq |
| have hm : m ≤ d := by simpa using Fintype.card_le_of_injective j hj |
| have hdim := hilbert_natDegree_eq_trdeg I (coordinateHilbert I) (coordinateHilbert_eventually I) |
| rw [ht] at hdim |
| have hh : affineDimension I = m+1 := by exact_mod_cast hdim |
| omega |
|
|
| theorem polynomial_constant_at_point {n : ℕ} (I : Ideal (MvPolynomial (Fin n) ℂ)) |
| (q : MvPolynomial (Fin n) ℂ) (t : ℂ) |
| (ht : algebraMap ℂ (MvPolynomial (Fin n) ℂ ⧸ I) t = Ideal.Quotient.mk I q) |
| (z : Fin n → ℤ) (hz : PointOn I z) : MvPolynomial.aeval (fun j => (z j:ℂ)) q = t := by |
| have hh := congrArg (pointEvaluation I z hz) ht |
| rw [AlgHom.commutes] at hh |
| change t = MvPolynomial.aeval (fun j => (z j:ℂ)) q at hh |
| exact hh.symm |
|
|
| theorem algebraic_polynomial_fibre_count {n : ℕ} (S : Finset (Fin n → ℤ)) (F : ℕ) |
| (q : MvPolynomial (Fin n) ℂ) |
| (hf : ∀ t : ℂ, (S.filter (fun z => MvPolynomial.aeval (fun j => (z j:ℂ)) q = t)).card ≤ F) |
| (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime] |
| (hq : IsAlgebraic ℂ (Ideal.Quotient.mk I q)) : (S.filter (PointOn I)).card ≤ F := by |
| obtain ⟨t,ht⟩ := algebraic_coordinate_constant (Ideal.Quotient.mk I q) hq |
| apply (Finset.card_le_card ?_).trans (hf t) |
| intro z hz |
| have h := Finset.mem_filter.mp hz |
| exact Finset.mem_filter.mpr ⟨h.1,polynomial_constant_at_point I q t ht z h.2⟩ |
|
|
| theorem algebraic_left_fibre_count {n d : ℕ} (S : Finset (Fin n → ℤ)) (F : ℕ) |
| (x : Fin d → Fin n) |
| (hf : ∀ c : Fin d → ℤ, (S.filter (fun z => ∀ i, z (x i) = c i)).card ≤ F) |
| (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime] |
| (hx : ∀ i, IsAlgebraic ℂ (Ideal.Quotient.mk I (MvPolynomial.X (x i)))) : |
| (S.filter (PointOn I)).card ≤ F := by |
| by_cases he : (S.filter (PointOn I)).Nonempty |
| · obtain ⟨z,hz⟩ := he |
| have hzI := (Finset.mem_filter.mp hz).2 |
| apply (Finset.card_le_card ?_).trans (hf (fun i => z (x i))) |
| intro w hw |
| obtain ⟨hwS,hwI⟩ := Finset.mem_filter.mp hw |
| refine Finset.mem_filter.mpr ⟨hwS,fun i => ?_⟩ |
| obtain ⟨t,ht⟩ := algebraic_coordinate_constant (Ideal.Quotient.mk I (MvPolynomial.X (x i))) (hx i) |
| exact Int.cast_injective (α := ℂ) ((coordinate_constant_at_point I w hwI (x i) t ht).trans |
| (coordinate_constant_at_point I z hzI (x i) t ht).symm) |
| · rw [Finset.not_nonempty_iff_eq_empty.mp he,Finset.card_empty] |
| exact Nat.zero_le _ |
|
|
| end |
| section |
| open MvPolynomial |
|
|
| theorem norm_generic_threshold_uniform {n d h : ℕ} (hd : 5 ≤ d) (hd' : d ≤ 8) |
| (hh : 2 ≤ h) (hh' : h ≤ d) (E D : ℕ) (hD : 0 < D) : |
| ∃ C : ℝ, 0 ≤ C ∧ |
| ∀ (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime] |
| (a b : Fin d → MvPolynomial (Fin n) ℂ) (q : MvPolynomial (Fin n) ℂ) |
| (θ : Fin d → ℂ), Function.Injective θ → |
| (∀ i, Ideal.Quotient.mk I (a i) ≠ 0) → |
| (∀ i, Ideal.Quotient.mk I (a i)^(d-2)*Ideal.Quotient.mk I (b i) = |
| Ideal.Quotient.mk I q-algebraMap ℂ _ (θ i)) → |
| Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤ → |
| Transcendental ℂ (Ideal.Quotient.mk I q) → affineDimension I = h → affineDegree I ≤ E → |
| (∀ i, (a i).totalDegree ≤ D) → q.totalDegree ≤ D → |
| ∀ (w : Fin n → ℕ), (∀ i, w i ≠ 0) → |
| (∀ i, a i ∈ weightedSpace w 5) → (∀ i, b i ∈ weightedSpace w 6) → |
| ∀ (ω : Fin n → ℝ), (∀ i, 0 < ω i) → (∀ i, (w i:ℝ) = 15*ω i) → |
| ∀ T : ℝ, 1 ≤ T → |
| T^h/((h.factorial:ℝ)*(normBoxEnvelope d h)^h)-C*T^(h-1) ≤ weightedHilbert I ω T := by |
| have hsucc : h-1+1 = h := by omega |
| obtain ⟨C,hC,hbound⟩ := norm_real_hilbert_uniform (n := n) (d := d) (h := h-1) (K := ℂ) |
| (k := d-2) (by omega) 5 6 (by norm_num) (by norm_num) E D hD 15 (by norm_num) |
| refine ⟨C,hC,fun I _ a b q θ hθ ha heq hg hq hdim hdeg hadeg hqdeg w hw haw hbw ω hω hscale T hT => ?_⟩ |
| have hb := hbound I a b q θ hθ ha heq (quotient_pair_generates a b hg I) hq |
| (hsucc ▸ hdim) (hsucc ▸ hh') hdeg hadeg hqdeg w hw haw hbw ω hω hscale T hT |
| rw [hsucc] at hb |
| norm_num only [Nat.min_def,show (5:ℕ) ≤ 6 by omega,ite_true,Nat.cast_ofNat] at hb |
| have hc := mul_le_mul_of_nonneg_right (normBox_generic_threshold d h hd hd' hh hh') |
| (pow_nonneg (by linarith : 0 ≤ T) h) |
| have hc' : T^h/((h.factorial:ℝ)*(normBoxEnvelope d h)^h) ≤ |
| 5*(((d-2)*5+6:ℕ):ℝ)^(h-1)*15^h/((h.factorial:ℝ)*5^h*6^h)*T^h := by |
| simpa only [one_div_mul_eq_div] using hc |
| linarith only [hb,hc'] |
| |
| theorem norm_full_threshold_uniform {n d : ℕ} (hd : 5 ≤ d) (hd' : d ≤ 8) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (I : Ideal (MvPolynomial (Fin n) ℂ)) [I.IsPrime] |
| (a b : Fin d → MvPolynomial (Fin n) ℂ) (q : MvPolynomial (Fin n) ℂ) |
| (θ : Fin d → ℂ), |
| (∀ i, Ideal.Quotient.mk I (a i) ≠ 0) → |
| (∀ i, Ideal.Quotient.mk I (a i)^(d-2)*Ideal.Quotient.mk I (b i) = |
| Ideal.Quotient.mk I q-algebraMap ℂ _ (θ i)) → |
| Algebra.adjoin ℂ (Set.range a ∪ Set.range b) = ⊤ → |
| Transcendental ℂ (Ideal.Quotient.mk I q) → affineDimension I = d+1 → |
| ∀ (w : Fin n → ℕ), (∀ i, w i ≠ 0) → |
| (∀ i, a i ∈ weightedSpace w 5) → (∀ i, b i ∈ weightedSpace w 6) → |
| ∀ (ω : Fin n → ℝ), (∀ i, 0 < ω i) → (∀ i, (w i:ℝ) = 15*ω i) → |
| ∀ T : ℝ, 1 ≤ T → |
| T^(d+1)/(((d+1).factorial:ℝ)*(normBoxEnvelope d (d+1))^(d+1))-C*T^d ≤ weightedHilbert I ω T := by |
| have hsucc : d-1+1 = d := by omega |
| have hsucc' : d-1+2 = d+1 := by omega |
| have hb := norm_full_real_hilbert_uniform (N := n) (n := d-1) (K := ℂ) |
| (d-2) 5 6 (by norm_num) (by norm_num) 15 (by norm_num) |
| rw [hsucc,hsucc'] at hb |
| obtain ⟨C,hC,hbound⟩ := hb |
| refine ⟨C,hC,fun I _ a b q θ ha heq hg hq hdim w hw haw hbw ω hω hscale T hT => ?_⟩ |
| have hb := hbound I a b q θ ha heq (quotient_pair_generates a b hg I) hq |
| hdim w hw haw hbw ω hω hscale T hT |
| have hc := mul_le_mul_of_nonneg_right (normBox_full_threshold d hd hd') |
| (pow_nonneg (by linarith : 0 ≤ T) (d+1)) |
| have hc' : T^(d+1)/(((d+1).factorial:ℝ)*(normBoxEnvelope d (d+1))^(d+1)) ≤ |
| (((d-2)*5+6:ℕ):ℝ)^(d-1)*15^(d+1)/(((d+1).factorial:ℝ)*5^d*6^d)*T^(d+1) := by |
| simpa only [one_div_mul_eq_div] using hc |
| norm_num only [Nat.cast_ofNat] at hb |
| linarith only [hb,hc'] |
| |
| open MvPolynomial |
| open scoped BigOperators Classical |
| |
| structure NormBoxSystem (n d D : ℕ) (ω : Fin n → ℝ) (w : Fin n → ℕ) where |
| Carrier : Type |
| [field : Field Carrier] |
| [charZero : CharZero Carrier] |
| v : Fin d → Carrier |
| independent : LinearIndependent ℚ v |
| σ : Fin d → Carrier →+* ℂ |
| θ : Fin d → ℂ |
| distinct : Function.Injective θ |
| x : Fin d → Fin n |
| xweight : ∀ i, ω (x i) = 1/3 |
| b : Fin d → MvPolynomial (Fin n) ℂ |
| q : MvPolynomial (Fin n) ℂ |
| generates : Algebra.adjoin ℂ |
| (Set.range (fun i => coordinateLinearForm x (σ i ∘ v)) ∪ Set.range b) = ⊤ |
| leftDegree : ∀ i, (coordinateLinearForm x (σ i ∘ v)).totalDegree ≤ D |
| qDegree : q.totalDegree ≤ D |
| leftWeight : ∀ i, coordinateLinearForm x (σ i ∘ v) ∈ weightedSpace w 5 |
| rightWeight : ∀ i, b i ∈ weightedSpace w 6 |
| |
| attribute [instance] NormBoxSystem.field NormBoxSystem.charZero |
| |
| noncomputable def NormBoxSystem.a {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ} |
| (M : NormBoxSystem n d D ω w) (i : Fin d) : MvPolynomial (Fin n) ℂ := |
| coordinateLinearForm M.x (M.σ i ∘ M.v) |
| |
| noncomputable def NormBoxSystem.equation {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ} |
| (M : NormBoxSystem n d D ω w) (i : Fin d) : MvPolynomial (Fin n) ℂ := |
| M.a i^(d-2)*M.b i-M.q+C (M.θ i) |
| |
| noncomputable def NormBoxSystem.ideal {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ} |
| (M : NormBoxSystem n d D ω w) : Ideal (MvPolynomial (Fin n) ℂ) := |
| Ideal.span (Set.range M.equation) |
| |
| theorem NormBoxSystem.quotient_equations {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ} |
| (M : NormBoxSystem n d D ω w) (I : Ideal (MvPolynomial (Fin n) ℂ)) (hI : M.ideal ≤ I) : |
| ∀ i, Ideal.Quotient.mk I (M.a i)^(d-2)*Ideal.Quotient.mk I (M.b i) = |
| Ideal.Quotient.mk I M.q-algebraMap ℂ _ (M.θ i) := by |
| intro i |
| have hm : M.equation i ∈ I := hI (Ideal.subset_span ⟨i,rfl⟩) |
| have he := (Ideal.Quotient.eq_zero_iff_mem).mpr hm |
| simp only [NormBoxSystem.equation,map_add,map_sub,map_mul,map_pow] at he |
| change Ideal.Quotient.mk I (M.a i)^(d-2)*Ideal.Quotient.mk I (M.b i)- |
| Ideal.Quotient.mk I M.q+algebraMap ℂ (MvPolynomial (Fin n) ℂ ⧸ I) (M.θ i) = 0 at he |
| linear_combination he |
| |
| theorem NormBoxSystem.quotient_left_nonzero {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ} |
| (M : NormBoxSystem n d D ω w) (I : Ideal (MvPolynomial (Fin n) ℂ)) |
| (z : Fin n → ℤ) (hz : PointOn I z) |
| (ha : ∀ i, MvPolynomial.aeval (fun j => (z j:ℂ)) (M.a i) ≠ 0) : |
| ∀ i, Ideal.Quotient.mk I (M.a i) ≠ 0 := by |
| intro i hi |
| have hh := congrArg (pointEvaluation I z hz) hi |
| rw [map_zero] at hh |
| change MvPolynomial.aeval (fun j => (z j:ℂ)) (M.a i) = 0 at hh |
| exact ha i hh |
| |
| theorem NormBoxSystem.right_real_weight {n d D : ℕ} {ω : Fin n → ℝ} {w : Fin n → ℕ} |
| (M : NormBoxSystem n d D ω w) (hω : ∀ i, 0 < ω i) (hscale : ∀ i, (w i:ℝ) = 15*ω i) |
| (I : Ideal (MvPolynomial (Fin n) ℂ)) (i : Fin d) : |
| Ideal.Quotient.mk I (M.b i) ∈ weightedFunctions I ω (2/5) := by |
| have hh := weightedSpace_map_le_weightedFunctions I w ω hω 15 (by norm_num) hscale (2/5) (by norm_num) |
| norm_num only [show (15:ℝ)*(2/5) = 6 by norm_num,Nat.floor_natCast] at hh |
| exact hh ⟨M.b i,M.rightWeight i,rfl⟩ |
| |
| end |
| open MvPolynomial |
| open scoped BigOperators Classical |
| |
| theorem norm_component_alternatives {n d h D : ℕ} (hd : 5 ≤ d) (hd' : d ≤ 8) (hh : 1 ≤ h) (hh' : h ≤ d+1) (hD : 0 < D) |
| (ω : Fin n → ℝ) (hω : ∀ i, 0 < ω i) (hωW : ∀ i, ω i ≤ 2/5) |
| (w : Fin n → ℕ) (hw : ∀ i, w i ≠ 0) (hscale : ∀ i, (w i:ℝ) = 15*ω i) |
| (M : NormBoxSystem n d D ω w) (Cbox : ℝ) (hCbox : 0 < Cbox) |
| (F E : ℕ) (ε : ℝ) (hε : 0 < ε) : |
| ∃ K H : ℝ, 1 ≤ K ∧ 0 ≤ H ∧ ∀ X : ℝ, 2 ≤ X → ∀ S : Finset (Fin n → ℤ), |
| (∀ z ∈ S, ∀ j, |(z j:ℝ)| ≤ Cbox*X^(ω j)) → |
| (∀ t : ℂ, (S.filter (fun z => MvPolynomial.aeval (fun j => (z j:ℂ)) M.q = t)).card ≤ F) → |
| (∀ c : Fin d → ℤ, (S.filter (fun z => ∀ i, z (M.x i) = c i)).card ≤ F) → |
| ∀ (I : Ideal (MvPolynomial (Fin n) ℂ)), I.IsPrime → M.ideal ≤ I → |
| affineDimension I = h → affineDegree I ≤ E → |
| (∀ i, Ideal.Quotient.mk I (M.a i) ≠ 0) → |
| ((S.filter (PointOn I)).card:ℝ) ≤ K*X^ε ∨ |
| ∀ T : ℝ, 1 ≤ T → T^h/((h.factorial:ℝ)*(normBoxEnvelope d h)^h)-H*T^(h-1) ≤ weightedHilbert I ω T := by |
| classical |
| have hsmall (X : ℝ) (hX : 2 ≤ X) (S : Finset (Fin n → ℤ)) (I : Ideal (MvPolynomial (Fin n) ℂ)) |
| (K : ℝ) (hKF : (F:ℝ) ≤ K) (hK : 0 ≤ K) (hcard : (S.filter (PointOn I)).card ≤ F) : |
| ((S.filter (PointOn I)).card:ℝ) ≤ K*X^ε := by |
| have hx : 1 ≤ X^ε := Real.one_le_rpow (by linarith) hε.le |
| calc |
| _ ≤ (F:ℝ) := by exact_mod_cast hcard |
| _ ≤ K := hKF |
| _ ≤ K*X^ε := by nlinarith |
| by_cases h1 : h = 1 |
| · subst h |
| obtain ⟨B,H,hB,hH,hcurve⟩ := norm_curve_alternatives (n := n) (d := d) (by omega) |
| (d-2) (by omega) ω hω M.x (1/3) (2/5) (2/5) Cbox (by norm_num) (by norm_num) |
| (by norm_num) hCbox M.xweight hωW E ε hε |
| let K : ℝ := max 1 (max B F) |
| have hK : 1 ≤ K := le_max_left _ _ |
| have hKF : (F:ℝ) ≤ K := (le_max_right _ _).trans (le_max_right _ _) |
| have hKB : B ≤ K := (le_max_left _ _).trans (le_max_right _ _) |
| refine ⟨K,H,hK,hH,fun X hX S hbox hnF haF I hI hMI hdim hdeg ha => ?_⟩ |
| let : I.IsPrime := hI |
| by_cases hq : IsAlgebraic ℂ (Ideal.Quotient.mk I M.q) |
| · exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_polynomial_fibre_count S F M.q hnF I hq)) |
| by_cases hxa : ∀ i, IsAlgebraic ℂ (Ideal.Quotient.mk I (MvPolynomial.X (M.x i))) |
| · exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_left_fibre_count S F M.x haF I hxa)) |
| push Not at hxa |
| obtain ⟨s,hs⟩ := hxa |
| have hsS (z) (hz : z ∈ S.filter (PointOn I)) := (Finset.mem_filter.mp hz).1 |
| have hh := hcurve M.Carrier M.v M.independent M.σ M.θ M.distinct X hX |
| (S.filter (PointOn I)) (fun z hz j => hbox z (hsS z hz) j) I hI hdim.le hdeg |
| (fun z hz => (Finset.mem_filter.mp hz).2) M.b M.q |
| (M.right_real_weight hω hscale I) hq (M.quotient_equations I hMI) s hs |
| rcases hh with hc | hHilb |
| · left |
| exact hc.trans (mul_le_mul_of_nonneg_right hKB (Real.rpow_nonneg (by linarith) _)) |
| · right |
| intro T hT |
| have hb := hHilb T (by linarith) |
| have hc := mul_le_mul_of_nonneg_right (normBox_curve_threshold d hd hd') (by linarith : 0 ≤ T) norm_num only [pow_one,Nat.factorial_one,Nat.cast_one,one_mul,Nat.sub_self,pow_zero,mul_one] |
| have hc' : T/normBoxEnvelope d 1 ≤ min (2/(1/3:ℝ)) (max (1/(1/3:ℝ)) (((d-1)*(d-2-1):ℕ)/(2/5:ℝ)))*T := by |
| simpa only [one_div_mul_eq_div] using hc |
| linarith only [hb,hc'] |
| · have hh2 : 2 ≤ h := by omega |
| by_cases hhd : h ≤ d |
| · obtain ⟨H,hH,hHilb⟩ := norm_generic_threshold_uniform (n := n) hd hd' hh2 hhd E D hD |
| let K : ℝ := max 1 F |
| have hK : 1 ≤ K := le_max_left _ _ |
| have hKF : (F:ℝ) ≤ K := le_max_right _ _ |
| refine ⟨K,H,hK,hH,fun X hX S _ hnF _ I hI hMI hdim hdeg ha => ?_⟩ |
| let : I.IsPrime := hI |
| by_cases hq : IsAlgebraic ℂ (Ideal.Quotient.mk I M.q) |
| · exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_polynomial_fibre_count S F M.q hnF I hq)) |
| · exact Or.inr (hHilb I M.a M.b M.q M.θ M.distinct ha (M.quotient_equations I hMI) |
| M.generates hq hdim hdeg M.leftDegree M.qDegree w hw M.leftWeight M.rightWeight ω hω hscale) |
| · have he : h = d+1 := by omega |
| subst h |
| obtain ⟨H,hH,hHilb⟩ := norm_full_threshold_uniform (n := n) hd hd' |
| let K : ℝ := max 1 F |
| have hK : 1 ≤ K := le_max_left _ _ |
| have hKF : (F:ℝ) ≤ K := le_max_right _ _ |
| refine ⟨K,H,hK,hH,fun X hX S _ hnF _ I hI hMI hdim _ ha => ?_⟩ |
| let : I.IsPrime := hI |
| by_cases hq : IsAlgebraic ℂ (Ideal.Quotient.mk I M.q) |
| · exact Or.inl (hsmall X hX S I K hKF (by linarith) (algebraic_polynomial_fibre_count S F M.q hnF I hq)) |
| · right |
| simpa only [Nat.add_sub_cancel] using hHilb I M.a M.b M.q M.θ ha (M.quotient_equations I hMI) |
| M.generates hq hdim w hw M.leftWeight M.rightWeight ω hω hscale |
|
|
| end QuarticPowerFree |
|
|
| end OAI |
|
|
