openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /HilbertDenominatorReduction.lean
| import OAI.NumberTheory.PiExponent.Analysis.ColonSeriesTelescope | |
| namespace OAI | |
| namespace PiExponentJets.W64 | |
| open scoped BigOperators | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k σ : Type*} [Field k] [Finite σ] | |
| noncomputable def colonHilbertNumerator (P : ℕ → Polynomial ℤ) (m : ℕ) : Polynomial ℤ := | |
| (∑ j ∈ Finset.range m, Polynomial.X ^ j * P j) * (1 - Polynomial.X) + | |
| Polynomial.X ^ m * P m | |
| theorem stabilized_colon_polynomial_numerator | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| (f : MvPolynomial σ k) (hf : f.IsHomogeneous 1) (m r : ℕ) | |
| (hstable : iteratedColon I f (m+1) = iteratedColon I f m) | |
| (P : ℕ → Polynomial ℤ) | |
| (hP : ∀ j : ℕ, sectionHilbertSeries (Ideal.span {f} ⊔ iteratedColon I f j) = | |
| (P j : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ r).val) : | |
| sectionHilbertSeries I = (colonHilbertNumerator P m : PowerSeries ℤ) * | |
| (PowerSeries.invOneSubPow ℤ (r+1)).val := by | |
| have hu : (PowerSeries.invOneSubPow ℤ r).val = | |
| (1-PowerSeries.X) * (PowerSeries.invOneSubPow ℤ (r+1)).val := by | |
| symm | |
| simpa only [pow_one, Nat.add_comm] using | |
| PowerSeries.one_sub_pow_mul_invOneSubPow_val_add_eq_invOneSubPow_val ℤ r 1 | |
| have hv : (PowerSeries.invOneSubPow ℤ r).val * (PowerSeries.invOneSubPow ℤ 1).val = | |
| (PowerSeries.invOneSubPow ℤ (r+1)).val := by | |
| rw [← Units.val_mul, ← PowerSeries.invOneSubPow_add] | |
| rw [stabilized_iteratedColon_series I hI f hf m hstable] | |
| simp_rw [hP] | |
| have hsum : (∑ j ∈ Finset.range m, | |
| PowerSeries.X ^ j * ((P j : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ r).val)) = | |
| (∑ j ∈ Finset.range m, PowerSeries.X ^ j * (P j : PowerSeries ℤ)) * | |
| (PowerSeries.invOneSubPow ℤ r).val := by | |
| simp only [Finset.sum_mul, mul_assoc] | |
| rw [hsum, mul_assoc (P m : PowerSeries ℤ), hv, hu] | |
| have hcoe : (colonHilbertNumerator P m : PowerSeries ℤ) = | |
| (∑ j ∈ Finset.range m, PowerSeries.X ^ j * (P j : PowerSeries ℤ)) * | |
| (1-PowerSeries.X) + PowerSeries.X ^ m * (P m : PowerSeries ℤ) := by | |
| change Polynomial.coeToPowerSeries.ringHom | |
| ((∑ j ∈ Finset.range m, Polynomial.X ^ j * P j) * (1-Polynomial.X) + | |
| Polynomial.X ^ m * P m) = _ | |
| rw [map_add, map_mul, map_sum] | |
| simp only [map_mul, map_sub, map_pow, map_one] | |
| simp only [Polynomial.coeToPowerSeries.ringHom_apply, Polynomial.coe_X] | |
| rw [hcoe] | |
| ring | |
| end PiExponentJets.W64 | |
| end OAI | |