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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 | 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
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