openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /HomogeneousHilbertExistence.lean
| import Mathlib.Data.Fintype.Option | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertDenominatorReduction | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertZeroVariables | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.StabilizedColonPowers | |
| import OAI.NumberTheory.PiExponent.Polynomials.CoordinateQuotientGrading | |
| namespace OAI | |
| namespace PiExponentJets.W64 | |
| universe u v | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k : Type v} [Field k] | |
| theorem iteratedColon_eq_colon_power {σ : Type u} [Finite σ] | |
| (I : Ideal (MvPolynomial σ k)) (f : MvPolynomial σ k) (n : ℕ) : | |
| iteratedColon I f n = I.colon {f^n} := by | |
| induction n with | |
| | zero => simp only [iteratedColon_zero, PiExponentJets.W06.colon_pow_zero] | |
| | succ n ih => rw [iteratedColon_succ, ih, PiExponentJets.W06.colon_pow_succ] | |
| theorem sectionHilbertSeries_rename {σ τ : Type u} [Finite σ] [Finite τ] | |
| (e : σ ≃ τ) (I : Ideal (MvPolynomial σ k)) : | |
| sectionHilbertSeries I = | |
| sectionHilbertSeries (I.map (MvPolynomial.rename e).toRingHom) := by | |
| ext n | |
| simp only [coeff_sectionHilbertSeries, | |
| PiExponentSiegel.W08.renameQuotientSection_finrank e I n] | |
| theorem sectionHilbertSeries_coordinate {σ : Type u} [Finite σ] | |
| (I : Ideal (MvPolynomial (Option σ) k)) (hX : MvPolynomial.X none ∈ I) : | |
| sectionHilbertSeries I = | |
| sectionHilbertSeries (PiExponentSiegel.W08.coordinateImageIdeal I) := by | |
| ext n | |
| simp only [coeff_sectionHilbertSeries, | |
| PiExponentSiegel.W08.coordinateQuotientSection_finrank I hX n] | |
| theorem exists_homogeneous_hilbert_numerator | |
| (σ : Type u) [Fintype σ] (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| ∃ P : Polynomial ℤ, sectionHilbertSeries I = | |
| (P : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card σ)).val := by | |
| classical | |
| refine Fintype.induction_empty_option | |
| (P := fun τ _ => ∀ J : Ideal (MvPolynomial τ k), | |
| J.IsHomogeneous (MvPolynomial.homogeneousSubmodule τ k) → | |
| ∃ P : Polynomial ℤ, sectionHilbertSeries J = | |
| (P : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card τ)).val) | |
| ?_ ?_ ?_ σ I hI | |
| · intro α β _ e ih J hJ | |
| let : Fintype α := Fintype.ofEquiv β e.symm | |
| obtain ⟨P, hP⟩ := ih (J.map (MvPolynomial.rename e.symm).toRingHom) | |
| (PiExponentSiegel.W08.renameImageIdeal_homogeneous e.symm J hJ) | |
| refine ⟨P, ?_⟩ | |
| rw [sectionHilbertSeries_rename e.symm J] | |
| simpa only [Fintype.card_congr e] using hP | |
| · intro J _ | |
| exact exists_hilbert_numerator_no_variables (by simp) J | |
| · intro α _ ih J hJ | |
| let f : MvPolynomial (Option α) k := MvPolynomial.X none | |
| have hf : f.IsHomogeneous 1 := MvPolynomial.isHomogeneous_X k none | |
| obtain ⟨m, hm⟩ := PiExponentJets.W06.exists_colon_pow_stable J f | |
| have hstable : iteratedColon J f (m+1) = iteratedColon J f m := by | |
| simpa only [iteratedColon_eq_colon_power] using hm | |
| let Jlower : ℕ → Ideal (MvPolynomial (Option α) k) := | |
| fun j => Ideal.span {f} ⊔ iteratedColon J f j | |
| have hlower : ∀ j, (Jlower j).IsHomogeneous | |
| (MvPolynomial.homogeneousSubmodule (Option α) k) := by | |
| intro j | |
| apply Ideal.IsHomogeneous.sup | |
| · apply Ideal.homogeneous_span (MvPolynomial.homogeneousSubmodule (Option α) k) | |
| intro g hg | |
| obtain rfl := Set.mem_singleton_iff.mp hg | |
| exact ⟨1, hf⟩ | |
| · exact iteratedColon_homogeneous J hJ f hf j | |
| have hX : ∀ j, MvPolynomial.X none ∈ Jlower j := by | |
| intro j | |
| have hinc : Ideal.span {f} ≤ Jlower j := le_sup_left | |
| exact hinc (Ideal.subset_span (Set.mem_singleton f)) | |
| choose P hP using fun j => ih (PiExponentSiegel.W08.coordinateImageIdeal (Jlower j)) | |
| (PiExponentSiegel.W08.coordinateImageIdeal_homogeneous (Jlower j) (hlower j)) | |
| have hseries : ∀ j, sectionHilbertSeries (Ideal.span {f} ⊔ iteratedColon J f j) = | |
| (P j : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card α)).val := by | |
| intro j | |
| rw [show Ideal.span {f} ⊔ iteratedColon J f j = Jlower j from rfl, | |
| sectionHilbertSeries_coordinate (Jlower j) (hX j)] | |
| exact hP j | |
| refine ⟨colonHilbertNumerator P m, ?_⟩ | |
| simpa only [Fintype.card_option] using | |
| stabilized_colon_polynomial_numerator J hJ f hf m (Fintype.card α) hstable P hseries | |
| end PiExponentJets.W64 | |
| end OAI | |