openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /HomogeneousHilbertPolynomial.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PositiveDimensionalHilbert | |
| import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousHilbertExistence | |
| namespace OAI | |
| namespace PiExponentJets.W64 | |
| universe u v | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k : Type v} [Field k] | |
| theorem rational_hilbert_numerator {σ : Type u} [Fintype σ] | |
| (I : Ideal (MvPolynomial σ k)) (P : Polynomial ℤ) | |
| (hP : sectionHilbertSeries I = | |
| (P : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card σ)).val) : | |
| rationalSectionHilbertSeries I = | |
| ((P.map (Int.castRingHom ℚ)) : PowerSeries ℚ) * | |
| (PowerSeries.invOneSubPow ℚ (Fintype.card σ)).val := by | |
| rw [rationalSectionHilbertSeries, hP, map_mul, map_int_invOneSubPow, | |
| ← Polynomial.polynomial_map_coe] | |
| theorem exists_homogeneous_hilbert_polynomial | |
| (σ : Type u) [Fintype σ] (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| ∃ p : Polynomial ℚ, ∃ N : ℕ, ∀ n : ℕ, N < n → | |
| (Module.finrank k (quotientSection I n) : ℚ) = p.eval (n : ℚ) := by | |
| obtain ⟨P, hP⟩ := exists_homogeneous_hilbert_numerator σ I hI | |
| refine ⟨Polynomial.hilbertPoly (P.map (Int.castRingHom ℚ)) (Fintype.card σ), | |
| (P.map (Int.castRingHom ℚ)).natDegree, ?_⟩ | |
| intro n hn | |
| rw [← coeff_rationalSectionHilbertSeries, rational_hilbert_numerator I P hP] | |
| exact Polynomial.coeff_mul_invOneSubPow_eq_hilbertPoly_eval _ hn | |
| theorem exists_homogeneous_hilbert_polynomial_degree_le | |
| (σ : Type u) [Fintype σ] (hσ : 0 < Fintype.card σ) | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| ∃ p : Polynomial ℚ, | |
| (∃ N : ℕ, ∀ n : ℕ, N < n → | |
| (Module.finrank k (quotientSection I n) : ℚ) = p.eval (n : ℚ)) ∧ | |
| p.natDegree ≤ Fintype.card σ - 1 := by | |
| obtain ⟨P, hP⟩ := exists_homogeneous_hilbert_numerator σ I hI | |
| refine ⟨Polynomial.hilbertPoly (P.map (Int.castRingHom ℚ)) (Fintype.card σ), | |
| ⟨(P.map (Int.castRingHom ℚ)).natDegree, ?_⟩, | |
| hilbertPolynomial_natDegree_le _ _ hσ⟩ | |
| intro n hn | |
| rw [← coeff_rationalSectionHilbertSeries, rational_hilbert_numerator I P hP] | |
| exact Polynomial.coeff_mul_invOneSubPow_eq_hilbertPoly_eval _ hn | |
| end PiExponentJets.W64 | |
| end OAI | |