import OAI.Algebra.Determinantal.Transcendence namespace OAI namespace PermanentBorder.SliceParameters open MvPolynomial lemma singularSystem_map {K L β : Type*} [CommRing K] [CommRing L] (φ : K →+* L) (f : MvPolynomial β K) (j : Option β) : map φ (singularSystem f j) = singularSystem (map φ f) j := by cases j with | none => rfl | some i => exact pderiv_map.symm noncomputable def universalSection {σ α β : Type*} [Fintype β] (F : MvPolynomial α (MvPolynomial σ ℤ)) : MvPolynomial β (MvPolynomial (σ ⊕ (α × β)) ℤ) := linearSection (fun i => X (Sum.inr i)) (map (rename (R := ℤ) (fun i : σ => (Sum.inl i : σ ⊕ (α × β)))).toRingHom F) lemma universalSection_specialize {K σ α β : Type*} [CommRing K] [Fintype β] (F : MvPolynomial α (MvPolynomial σ ℤ)) (θ : σ → K) (H : α × β → K) : map (eval₂Hom (Int.castRingHom K) (Sum.elim θ H)) (universalSection F) = linearSection H (map (eval₂Hom (Int.castRingHom K) θ) F) := by rw [universalSection, linearSection_map, map_map] have he : (eval₂Hom (Int.castRingHom K) (Sum.elim θ H)).comp (rename (R := ℤ) (fun i : σ => (Sum.inl i : σ ⊕ (α × β)))).toRingHom = eval₂Hom (Int.castRingHom K) θ := by ext i <;> simp rw [he] have hH : (fun i : α × β => eval₂Hom (Int.castRingHom K) (Sum.elim θ H) (X (Sum.inr i))) = H := by funext i exact eval₂_X _ _ _ rw [hH] lemma universalSection_macaulay_map {M σ α β : Type*} [Field M] [Fintype β] (F : MvPolynomial α (MvPolynomial σ ℤ)) (r D : ℕ) (t : σ → M) (A : α × β → M) : (macaulayMatrix (singularDegrees r) (singularSystem (universalSection (β := β) F)) D).map (eval₂Hom (Int.castRingHom M) (Sum.elim t A)) = macaulayMatrix (singularDegrees r) (singularSystem (linearSection A (map (eval₂Hom (Int.castRingHom M) t) F))) D := by rw [← macaulayMatrix_map] apply congrArg (fun Q => macaulayMatrix (singularDegrees r) Q D) funext j rw [singularSystem_map, universalSection_specialize] theorem transfer_section_certificate {K L σ α β : Type*} [Field K] [Field L] [CharZero L] [Infinite L] [Fintype β] (F : MvPolynomial α (MvPolynomial σ ℤ)) (r D : ℕ) (θ : σ → K) (H : α × β → K) (h : Function.Surjective (macaulayMatrix (singularDegrees r) (singularSystem (linearSection H (map (eval₂Hom (Int.castRingHom K) θ) F))) D).mulVec) : ∃ θ' : σ → L, ∃ H' : α × β → L, Function.Surjective (macaulayMatrix (singularDegrees r) (singularSystem (linearSection H' (map (eval₂Hom (Int.castRingHom L) θ') F))) D).mulVec := by classical let P := macaulayMatrix (singularDegrees r) (singularSystem (universalSection (β := β) F)) D have hp : Function.Surjective (P.map (eval₂Hom (Int.castRingHom K) (Sum.elim θ H))).mulVec := by dsimp only [P] rwa [universalSection_macaulay_map] obtain ⟨w,hw⟩ := integer_polynomial_rank_transfer (L := L) P (Sum.elim θ H) hp refine ⟨fun i => w (Sum.inl i), fun i => w (Sum.inr i), ?_⟩ have hw' : Sum.elim (fun i => w (Sum.inl i)) (fun i => w (Sum.inr i)) = w := by funext i cases i <;> rfl dsimp only [P] at hw rw [← universalSection_macaulay_map, hw'] exact hw end PermanentBorder.SliceParameters namespace PermanentBorder.SliceParameters open MvPolynomial lemma macaulay_no_nonzero_zero {K σ ι : Type*} [Field K] [Fintype σ] [Fintype ι] (e : ι → ℤ) (F : ι → MvPolynomial σ K) (hF : ∀ i, (F i).IsWeightedHomogeneous (fun _ : σ => (1 : ℤ)) (e i)) (D : ℕ) (h : Function.Surjective (macaulayMatrix e F D).mulVec) (y : σ → K) (hy : ∀ i, eval y (F i) = 0) : y = 0 := by classical have hz : y ∈ zeroLocus K (Ideal.span (Set.range F)) := by rw [zeroLocus_span] rintro _ ⟨i,rfl⟩ exact hy i have hD := (macaulay_surjective_iff e F hF D).mp h ext j have hx : (X j : MvPolynomial σ K)^D ∈ Ideal.span (Set.range F) := hD _ (homogeneous_intWeight (by simpa only [one_mul] using (isHomogeneous_X K j).pow D)) have hv : y j ^ D = 0 := by simpa using hz _ hx by_contra hj exact pow_ne_zero D hj hv theorem certificate_smooth_equations {K β : Type*} [Field K] [Fintype β] (g : MvPolynomial β K) (r D : ℕ) (hg : g.IsHomogeneous r) (h : Function.Surjective (macaulayMatrix (singularDegrees r) (singularSystem g) D).mulVec) : ∀ y : β → K, eval y g = 0 → (∀ j, eval y (pderiv j g) = 0) → y = 0 := by intro y hy hd apply macaulay_no_nonzero_zero _ _ (singularSystem_homogeneous hg) D h y intro j cases j with | none => exact hy | some j => exact hd j end PermanentBorder.SliceParameters namespace PermanentBorder.SliceParameters open MvPolynomial universe u v variable (K : Type u) [Field K] (ι : Type v) abbrev GenericClosed := AlgebraicClosure (FractionRing (MvPolynomial ι K)) noncomputable def independentParameters : ι → GenericClosed K ι := fun i => algebraMap (FractionRing (MvPolynomial ι K)) (GenericClosed K ι) (algebraMap (MvPolynomial ι K) (FractionRing (MvPolynomial ι K)) (X i)) lemma independentParameters_independent : AlgebraicIndependent K (independentParameters K ι) := by let R := MvPolynomial ι K let Q := FractionRing R have hQ : AlgebraicIndependent K (fun i : ι => algebraMap R Q (X i)) := (MvPolynomial.algebraicIndependent_X ι K).map' (f := IsScalarTower.toAlgHom K R Q) (IsFractionRing.injective R Q) exact hQ.map' (f := IsScalarTower.toAlgHom K Q (GenericClosed K ι)) (RingHom.injective (algebraMap Q (GenericClosed K ι))) end PermanentBorder.SliceParameters namespace PermanentBorder.SliceParameters open MvPolynomial universe u v lemma specialize_tower {K L σ α : Type*} [CommRing K] [CommRing L] (φ : K →+* L) (F : MvPolynomial α (MvPolynomial σ ℤ)) (θ : σ → K) : map φ (map (eval₂Hom (Int.castRingHom K) θ) F) = map (eval₂Hom (Int.castRingHom L) (fun i => φ (θ i))) F := by rw [map_map] apply congrArg (fun ψ : MvPolynomial σ ℤ →+* L => map ψ F) ext i <;> simp theorem exists_complex_smooth_section {K : Type u} [Field K] {σ : Type v} {v d n r : ℕ} (F : MvPolynomial (Fin (v+1)) (MvPolynomial σ ℤ)) (hF : F.IsHomogeneous r) (θ : σ → K) (hf : map (eval₂Hom (Int.castRingHom K) θ) F ≠ 0) (hdim : ringKrullDim (MvPolynomial (Fin (v+1)) K ⧸ Ideal.span (Set.range (fun i => pderiv i (map (eval₂Hom (Int.castRingHom K) θ) F)))) ≤ n) (hn : n+d < v+1) : ∃ θ' : σ → ℂ, ∃ H' : Fin (v+1) × Fin (d+1) → ℂ, let g := linearSection H' (map (eval₂Hom (Int.castRingHom ℂ) θ') F) g ≠ 0 ∧ g.IsHomogeneous r ∧ (∀ y : Fin (d+1) → ℂ, eval y g = 0 → (∀ j, eval y (pderiv j g) = 0) → y = 0) := by classical let M := GenericClosed K (Fin (v+1) × Fin (d+1)) let H := independentParameters K (Fin (v+1) × Fin (d+1)) obtain ⟨D,hD⟩ := generic_slice_certificate (L := M) _ hf (hF.map _) hdim hn H (independentParameters_independent K _) rw [specialize_tower] at hD obtain ⟨t,A,hA⟩ := transfer_section_certificate (L := ℂ) F r D (fun i => algebraMap K M (θ i)) H hD refine ⟨t,A, ?_⟩ let g := linearSection A (map (eval₂Hom (Int.castRingHom ℂ) t) F) have hg : g.IsHomogeneous r := linearSection_homogeneous A (hF.map _) have hs := certificate_smooth_equations g r D hg hA refine ⟨?_,hg,hs⟩ intro h change g = 0 at h have hzero := hs (fun _ => 1) (by rw [h, map_zero]) (by intro j rw [h, map_zero, map_zero]) exact one_ne_zero (congrFun hzero (0 : Fin (d+1))) end PermanentBorder.SliceParameters namespace PermanentBorder.SliceParameters open MvPolynomial lemma gradientIdeal_rename {K α β : Type*} [CommRing K] (e : α ≃ β) (f : MvPolynomial α K) : Ideal.span (Set.range (fun j => pderiv j (rename e f))) = (Ideal.span (Set.range (fun i => pderiv i f))).map (renameEquiv K e).toRingEquiv.toRingHom := by rw [Ideal.map_span] congr 1 ext p constructor · rintro ⟨j,rfl⟩ refine ⟨pderiv (e.symm j) f,⟨e.symm j,rfl⟩,?_⟩ change rename e (pderiv (e.symm j) f) = pderiv j (rename e f) rw [← pderiv_rename e.injective,e.apply_symm_apply] · rintro ⟨_,⟨i,rfl⟩,rfl⟩ exact ⟨e i,pderiv_rename e.injective i f⟩ lemma gradientDimension_rename {K α β : Type*} [CommRing K] (e : α ≃ β) (f : MvPolynomial α K) : ringKrullDim (MvPolynomial β K ⧸ Ideal.span (Set.range (fun j => pderiv j (rename e f)))) = ringKrullDim (MvPolynomial α K ⧸ Ideal.span (Set.range (fun i => pderiv i f))) := by exact ringKrullDim_eq_of_ringEquiv (Ideal.quotientEquiv _ _ (renameEquiv K e).toRingEquiv (gradientIdeal_rename e f)).symm lemma nonzero_of_gradientDimension {K α : Type*} [Field K] [Fintype α] (f : MvPolynomial α K) (n : ℕ) (hdim : ringKrullDim (MvPolynomial α K ⧸ Ideal.span (Set.range (fun i => pderiv i f))) ≤ n) (hn : n < Fintype.card α) : f ≠ 0 := by intro h rw [h] at hdim have hz : Ideal.span (Set.range (fun i : α => pderiv i (0 : MvPolynomial α K))) = ⊥ := by apply le_antisymm _ bot_le apply Ideal.span_le.mpr rintro _ ⟨i,rfl⟩ simp rw [hz,ringKrullDim_eq_of_ringEquiv (RingEquiv.quotientBot _), MvPolynomial.ringKrullDim_of_isNoetherianRing] at hdim have hdim' : ((Fintype.card α : ℕ) : WithBot ℕ∞) ≤ n := by simpa using hdim have hi : Fintype.card α ≤ n := by exact_mod_cast hdim' omega theorem exists_complex_smooth_section_finite {K α σ : Type} [Field K] [Fintype α] {v d n r : ℕ} (hv : Fintype.card α = v+1) (F : MvPolynomial α (MvPolynomial σ ℤ)) (hF : F.IsHomogeneous r) (θ : σ → K) (hf : map (eval₂Hom (Int.castRingHom K) θ) F ≠ 0) (hdim : ringKrullDim (MvPolynomial α K ⧸ Ideal.span (Set.range (fun i => pderiv i (map (eval₂Hom (Int.castRingHom K) θ) F)))) ≤ n) (hn : n+d < v+1) : ∃ θ' : σ → ℂ, ∃ H' : α × Fin (d+1) → ℂ, let g := linearSection H' (map (eval₂Hom (Int.castRingHom ℂ) θ') F) g ≠ 0 ∧ g.IsHomogeneous r ∧ (∀ y : Fin (d+1) → ℂ, eval y g = 0 → (∀ j, eval y (pderiv j g) = 0) → y = 0) := by classical let e : α ≃ Fin (v+1) := Fintype.equivFinOfCardEq hv let F' := rename e F have he (L : Type) [CommRing L] (t : σ → L) : map (eval₂Hom (Int.castRingHom L) t) F' = rename e (map (eval₂Hom (Int.castRingHom L) t) F) := by exact map_rename _ _ _ have hf' : map (eval₂Hom (Int.castRingHom K) θ) F' ≠ 0 := by rw [he] exact fun h => hf ((rename_injective _ e.injective).eq_iff.mp (h.trans (map_zero _).symm)) have hd' : ringKrullDim (MvPolynomial (Fin (v+1)) K ⧸ Ideal.span (Set.range (fun i => pderiv i (map (eval₂Hom (Int.castRingHom K) θ) F')))) ≤ n := by rw [he,gradientDimension_rename] exact hdim obtain ⟨t,H,hh⟩ := exists_complex_smooth_section F' hF.rename_isHomogeneous θ hf' hd' hn refine ⟨t,(fun ij => H (e ij.1,ij.2)),?_⟩ have hl (f : MvPolynomial α ℂ) : linearSection H (rename e f) = linearSection (fun ij => H (e ij.1,ij.2)) f := by have hhom : (linearSection H).comp (rename e) = linearSection (fun ij => H (e ij.1,ij.2)) := by apply MvPolynomial.algHom_ext intro i simp only [AlgHom.comp_apply,rename_X,linearSection_X] rfl exact DFunLike.congr_fun hhom f rw [he,hl] at hh exact hh end PermanentBorder.SliceParameters namespace PermanentBorder.Parameters abbrev auxiliarySize (m : ℕ) := m / 11 abbrev selectorIndex (k : ℕ) := (k-1) / 3 abbrev selectorDegree (k : ℕ) := k - 2 * selectorIndex k abbrev defectThreshold (k : ℕ) := (selectorIndex k + 2).choose 2 abbrev codimension (k : ℕ) := defectThreshold k - 2*k abbrev sectionDimension (k : ℕ) := k^2 / 100 theorem size_bound {m : ℕ} (hm : 1408 ≤ m) : 128 ≤ auxiliarySize m ∧ (m : ℝ) / 12 ≤ auxiliarySize m ∧ 11 * auxiliarySize m ≤ m := by dsimp [auxiliarySize] have h : m ≤ 12 * (m / 11) := by omega refine ⟨by omega, ?_, by omega⟩ have hr : (m : ℝ) ≤ 12 * (m / 11 : ℕ) := by exact_mod_cast h linarith theorem index_bounds {k : ℕ} (hk : 128 ≤ k) : (k : ℝ) / 4 ≤ selectorIndex k ∧ selectorIndex k ≤ k / 2 ∧ (k : ℝ) / 4 ≤ (selectorDegree k - 1 : ℕ) ∧ 2 ≤ selectorDegree k := by have hq : k ≤ 4 * selectorIndex k := by dsimp [selectorIndex]; omega have hr : k ≤ 4 * (selectorDegree k - 1) := by dsimp [selectorDegree, selectorIndex]; omega refine ⟨?_, by dsimp [selectorIndex]; omega, ?_, ?_⟩ · have hc : (k : ℝ) ≤ 4 * selectorIndex k := by exact_mod_cast hq linarith · have hc : (k : ℝ) ≤ 4 * (selectorDegree k - 1 : ℕ) := by exact_mod_cast hr linarith · dsimp [selectorDegree, selectorIndex]; omega theorem defect_formula (k : ℕ) : (defectThreshold k : ℝ) = ((selectorIndex k : ℝ)+2) * ((selectorIndex k : ℝ)+1) / 2 := by simp only [defectThreshold, Nat.cast_choose_two, Nat.cast_add, Nat.cast_ofNat] ring theorem codimension_bounds {k : ℕ} (hk : 128 ≤ k) : (k : ℝ)^2 / 64 ≤ codimension k ∧ 1 ≤ codimension k ∧ codimension k ≤ k^2 := by have hkR : (128 : ℝ) ≤ k := by exact_mod_cast hk have hq := (index_bounds hk).1 have hq0 : (0 : ℝ) ≤ selectorIndex k := Nat.cast_nonneg _ have hqU : (selectorIndex k : ℝ) ≤ (k : ℝ)/3 := by have : 3 * selectorIndex k ≤ k := by dsimp [selectorIndex]; omega have hc : (3:ℝ)*selectorIndex k ≤ k := by exact_mod_cast this linarith have hT := defect_formula k have hlow : (k : ℝ)^2 / 32 ≤ defectThreshold k := by nlinarith [sq_nonneg ((selectorIndex k : ℝ) - (k : ℝ)/4)] have hdelta : (k : ℝ)^2/64 ≤ (defectThreshold k : ℝ) - 2*k := by nlinarith [mul_nonneg (show 0 ≤ (k : ℝ) from by positivity) (sub_nonneg.mpr hkR)] have hsub : 2*k ≤ defectThreshold k := by have : (2*k : ℝ) ≤ defectThreshold k := by nlinarith [sq_nonneg (k : ℝ)] exact_mod_cast this have hd : (codimension k : ℝ) = (defectThreshold k : ℝ) - 2*k := by rw [codimension, Nat.cast_sub hsub, Nat.cast_mul, Nat.cast_ofNat] have hupper : (defectThreshold k : ℝ) < (k : ℝ)^2 := by nlinarith [mul_nonpos_of_nonpos_of_nonneg (show (selectorIndex k : ℝ) - k/3 ≤ 0 by linarith) (show (0:ℝ) ≤ (selectorIndex k : ℝ) + k/3 + 3 by positivity)] refine ⟨by simpa only [hd] using hdelta, ?_, ?_⟩ · have : (1 : ℝ) ≤ codimension k := by rw [hd]; nlinarith [sq_nonneg ((k : ℝ)-128)] exact_mod_cast this · have : (codimension k : ℝ) ≤ (k : ℝ)^2 := by rw [hd]; nlinarith exact_mod_cast this theorem section_bounds {k : ℕ} (hk : 128 ≤ k) : 2 ≤ sectionDimension k ∧ sectionDimension k < k^2 ∧ sectionDimension k - 1 < codimension k ∧ (k : ℝ)^2 / 200 ≤ (sectionDimension k - 1 : ℕ) := by have hk2 : 16384 ≤ k^2 := by nlinarith have hd2 : 2 ≤ sectionDimension k := by dsimp [sectionDimension]; omega have hcast : (k : ℝ)^2 < 100*((sectionDimension k : ℝ)+1) := by have : k^2 < 100*(sectionDimension k+1) := by dsimp [sectionDimension]; omega exact_mod_cast this have hcast2 : 100*(sectionDimension k : ℝ) ≤ (k : ℝ)^2 := by have : 100*sectionDimension k ≤ k^2 := by dsimp [sectionDimension]; omega exact_mod_cast this have hkR : (128 : ℝ) ≤ k := by exact_mod_cast hk have hm : ((sectionDimension k - 1 : ℕ) : ℝ) = sectionDimension k - 1 := by rw [Nat.cast_sub (by omega), Nat.cast_one] refine ⟨hd2, by dsimp [sectionDimension]; omega, ?_, ?_⟩ · have hδ := (codimension_bounds hk).1 have : ((sectionDimension k - 1 : ℕ) : ℝ) < codimension k := by rw [hm] nlinarith [sq_nonneg (k : ℝ)] exact_mod_cast this · rw [hm] nlinarith [sq_nonneg ((k : ℝ)-128)] theorem explicit_constant {m : ℕ} (hm : 1408 ≤ m) : (m : ℝ)^3 / (5529600 * Real.exp 1) ≤ (((selectorDegree (auxiliarySize m) - 1 : ℕ) : ℝ) * ((sectionDimension (auxiliarySize m) - 1 : ℕ) : ℝ)) / (4 * Real.exp 1) := by let k := auxiliarySize m have hk : 128 ≤ k := (size_bound hm).1 have hmk : (m : ℝ)/12 ≤ k := (size_bound hm).2.1 have hr := (index_bounds hk).2.2.1 have hd := (section_bounds hk).2.2.2 have hprod : (k : ℝ)^3 / 800 ≤ ((selectorDegree k - 1 : ℕ) : ℝ) * ((sectionDimension k - 1 : ℕ) : ℝ) := by have hmul := mul_le_mul hr hd (by positivity : (0:ℝ) ≤ (k : ℝ)^2 / 200) (by positivity : (0:ℝ) ≤ ((selectorDegree k - 1 : ℕ) : ℝ)) nlinarith have hcubes : (m : ℝ)^3 ≤ 1728 * (k : ℝ)^3 := by have := pow_le_pow_left₀ (by positivity : (0:ℝ) ≤ (m:ℝ)) (show (m : ℝ) ≤ 12 * k by linarith) 3 nlinarith have he := Real.exp_pos (1 : ℝ) apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr dsimp [k] at hprod hcubes ⊢ nlinarith end PermanentBorder.Parameters namespace PermanentBorder open MvPolynomial lemma SliceParameters.linearSection_eq_affineSubstitution {α β : Type*} [Fintype β] (H : α × β → ℂ) : SliceParameters.linearSection H = affineSubstitution (fun _ => 0) (fun i j => H (i,j)) := by apply MvPolynomial.algHom_ext intro i simp [SliceParameters.linearSection,SliceParameters.sectionForm,affineSubstitution] theorem HasBorderDeterminant.selector_lower_of_critical_dimension {K : Type} [Field K] {m n k q d B : ℕ} (h : HasBorderDeterminant n (permanentPolynomial m)) (hk : 0 < k) (hq : q ≤ k) (hr : 2 ≤ k-2*q) (hd : 2 ≤ d) (hm : 11*k-2*q ≤ m) (θ : Fin k → K) (hdim : ringKrullDim (MvPolynomial (Variables k) K ⧸ Ideal.span (Set.range (fun i => pderiv i (Selector.diagonalCoefficient (Matrix.of fun a b => (X (a,b) : MvPolynomial (Variables k) K)) (fun i => C (θ i)) q)))) ≤ B) (hB : B+(d-1) < k^2) : (((k-2*q-1 : ℕ) : ℝ) * ((d-1 : ℕ) : ℝ)) / (4*Real.exp 1) ≤ n := by classical have hk2 : 0 < k^2 := pow_pos hk _ have hc : Fintype.card (Variables k) = k^2 := by simp [Variables,pow_two] have hv : Fintype.card (Variables k) = (k^2-1)+1 := by omega let F : MvPolynomial (Variables k) (MvPolynomial (Fin k) ℤ) := Selector.integralCoefficientFamily k q have hd' : ringKrullDim (MvPolynomial (Variables k) K ⧸ Ideal.span (Set.range (fun i => pderiv i (map (eval₂Hom (Int.castRingHom K) θ) F)))) ≤ B := by rw [show map (eval₂Hom (Int.castRingHom K) θ) F = Selector.diagonalCoefficient (Matrix.of fun a b => (X (a,b) : MvPolynomial (Variables k) K)) (fun i => C (θ i)) q from Selector.integralCoefficientFamily_specialize k q θ] exact hdim have hFne := SliceParameters.nonzero_of_gradientDimension (map (eval₂Hom (Int.castRingHom K) θ) F) B hd' (by omega : B < Fintype.card (Variables k)) obtain ⟨t,H,hs⟩ := SliceParameters.exists_complex_smooth_section_finite hv F (Selector.integralCoefficientFamily_homogeneous k q) θ hFne hd' (by omega : B+(d-1) < (k^2-1)+1) have hb := (h.coefficientPolynomial hq hm t).affineSubstitution (fun _ => 0) (fun i j => H (i,j)) rw [← SliceParameters.linearSection_eq_affineSubstitution, ← Selector.integralCoefficientFamily_complex] at hb have he := Polar.HasBorderDeterminant.smooth_lower_bound hb (by simpa only [Fintype.card_fin] using (show 2 ≤ d-1+1 by omega)) hr hs.2.1 hs.2.2 simpa only [Fintype.card_fin,Nat.add_sub_cancel] using he end PermanentBorder namespace PermanentBorder open MvPolynomial Parameters end PermanentBorder end OAI