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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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.NumberTheory.PiExponent.Analysis.RegularSequenceSeries
namespace OAI
namespace PiExponentJets.W64
open scoped BigOperators
attribute [local instance] MvPolynomial.gradedAlgebra
noncomputable def degreePolynomial (D : ℕ → ℕ) (m : ℕ) : Polynomial ℤ :=
∏ i ∈ Finset.range m, ∑ j ∈ Finset.range (D i), (Polynomial.X : Polynomial ℤ) ^ j
theorem degreePolynomial_eval_one (D : ℕ → ℕ) (m : ℕ) :
(degreePolynomial D m).eval 1 = ∏ i ∈ Finset.range m, (D i : ℤ) := by
simp [degreePolynomial, Polynomial.eval_prod, Polynomial.eval_finsetSum]
variable {k σ : Type*} [Field k] [Fintype σ]
theorem regular_sequence_series_polynomial
(hσ : 0 < Fintype.card σ)
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length = Fintype.card σ) :
sectionHilbertSeries (Ideal.ofList rs) =
((degreePolynomial (degreeAt degrees) rs.length) : PowerSeries ℤ) := by
rw [regular_sequence_series_geometric hσ rs degrees hhom hreg hlen]
change _ = (Polynomial.coeToPowerSeries.ringHom : Polynomial ℤ →+* PowerSeries ℤ)
(∏ i ∈ Finset.range rs.length, ∑ j ∈ Finset.range (degreeAt degrees i),
(Polynomial.X : Polynomial ℤ) ^ j)
rw [map_prod]
apply Finset.prod_congr rfl
intro i hi
rw [map_sum]
apply Finset.sum_congr rfl
intro j hj
rw [map_pow]
simp only [Polynomial.coeToPowerSeries.ringHom_apply, Polynomial.coe_X]
theorem regular_sequence_section_dimension_coeff
(hσ : 0 < Fintype.card σ)
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length = Fintype.card σ) (n : ℕ) :
(Module.finrank k (quotientSection (Ideal.ofList rs) n) : ℤ) =
(degreePolynomial (degreeAt degrees) rs.length).coeff n := by
have h := congrArg (PowerSeries.coeff n)
(regular_sequence_series_polynomial hσ rs degrees hhom hreg hlen)
simpa only [coeff_sectionHilbertSeries, Polynomial.coeff_coe] using h
theorem regular_sequence_section_dimension_eq_zero
(hσ : 0 < Fintype.card σ)
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length = Fintype.card σ) (n : ℕ)
(hn : n ∉ (degreePolynomial (degreeAt degrees) rs.length).support) :
Module.finrank k (quotientSection (Ideal.ofList rs) n) = 0 := by
have h := regular_sequence_section_dimension_coeff hσ rs degrees hhom hreg hlen n
have hz : (degreePolynomial (degreeAt degrees) rs.length).coeff n = 0 := by
simpa only [Polynomial.mem_support_iff, not_not] using hn
rw [hz] at h
exact_mod_cast h
theorem regular_sequence_section_dimension_sum
(hσ : 0 < Fintype.card σ)
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length = Fintype.card σ) :
(∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support,
Module.finrank k (quotientSection (Ideal.ofList rs) n)) =
∏ i ∈ Finset.range rs.length, degreeAt degrees i := by
have h :
(∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support,
(Module.finrank k (quotientSection (Ideal.ofList rs) n) : ℤ)) =
∏ i ∈ Finset.range rs.length, (degreeAt degrees i : ℤ) := by
calc
_ = ∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support,
(degreePolynomial (degreeAt degrees) rs.length).coeff n := by
apply Finset.sum_congr rfl
intro n _
exact regular_sequence_section_dimension_coeff hσ rs degrees hhom hreg hlen n
_ = (degreePolynomial (degreeAt degrees) rs.length).eval 1 := by
simp [Polynomial.eval_eq_sum, Polynomial.sum]
_ = _ := degreePolynomial_eval_one _ _
exact_mod_cast h
theorem regular_sequence_section_dimension_sum_le
(hσ : 0 < Fintype.card σ)
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length = Fintype.card σ) (D : ℕ)
(hD : ∀ i, degrees i ≤ D) :
(∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support,
Module.finrank k (quotientSection (Ideal.ofList rs) n)) ≤ D ^ rs.length := by
rw [regular_sequence_section_dimension_sum hσ rs degrees hhom hreg hlen]
calc
_ ≤ ∏ _i ∈ Finset.range rs.length, D := by
apply Finset.prod_le_prod
intro i hi
simpa only [degreeAt, dite_eq_left (Finset.mem_range.mp hi)] using
hD ⟨i, Finset.mem_range.mp hi⟩
_ = _ := by simp
end PiExponentJets.W64
end OAI
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