| import Mathlib.Algebra.Ring.GeomSum |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularPrefixes |
| import OAI.NumberTheory.PiExponent.Polynomials.HilbertPolynomialBase |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| open scoped BigOperators |
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| variable {k σ : Type*} [Field k] [Finite σ] |
|
|
| noncomputable def degreeAt {m : ℕ} (degrees : Fin m → ℕ) (i : ℕ) : ℕ := |
| if hi : i < m then degrees ⟨i, hi⟩ else 0 |
|
|
| theorem regular_prefix_series |
| (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) |
| (n : ℕ) (hn : n ≤ rs.length) : |
| sectionHilbertSeries (Ideal.ofList (rs.take n)) = |
| (∏ i ∈ Finset.range n, (1 - PowerSeries.X ^ degreeAt degrees i)) * |
| sectionHilbertSeries (⊥ : Ideal (MvPolynomial σ k)) := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have hlt : n < rs.length := hn |
| have hn' : n ≤ rs.length := Nat.le_of_lt hlt |
| rw [PiExponentSiegel.W20.prefixIdeal_succ rs ⟨n, hlt⟩] |
| simp only [Fin.getElem_fin] |
| have hh : (rs[n]'hlt).IsHomogeneous (degrees ⟨n, hlt⟩) := by |
| simpa only [Fin.getElem_fin] using hhom ⟨n, hlt⟩ |
| have hr : IsRightRegular (Ideal.Quotient.mk (Ideal.ofList (rs.take n)) (rs[n]'hlt)) := by |
| simpa only [Fin.getElem_fin] using |
| PiExponentSiegel.W20.regular_next_mod_prefix rs hreg ⟨n, hlt⟩ |
| rw [sectionHilbertSeries_regular_step _ |
| (PiExponentSiegel.W20.prefixIdeal_homogeneous rs degrees hhom n) |
| (rs[n]'hlt) hh hr] |
| rw [ih hn', Finset.prod_range_succ] |
| simp only [degreeAt, dite_eq_left hlt] |
| ac_rfl |
|
|
| theorem degree_factors_mul_invOneSubPow (D : ℕ → ℕ) (m : ℕ) : |
| (∏ i ∈ Finset.range m, (1 - (PowerSeries.X : PowerSeries ℤ) ^ D i)) * |
| (PowerSeries.invOneSubPow ℤ m).val = |
| ∏ i ∈ Finset.range m, ∑ j ∈ Finset.range (D i), |
| (PowerSeries.X : PowerSeries ℤ) ^ j := by |
| have hfactor : |
| (∏ i ∈ Finset.range m, (1 - (PowerSeries.X : PowerSeries ℤ) ^ D i)) = |
| (∏ i ∈ Finset.range m, ∑ j ∈ Finset.range (D i), |
| (PowerSeries.X : PowerSeries ℤ) ^ j) * (1 - PowerSeries.X) ^ m := by |
| simp_rw [← geom_sum_mul_neg] |
| rw [Finset.prod_mul_distrib] |
| simp |
| rw [hfactor, mul_assoc, ← PowerSeries.invOneSubPow_inv_eq_one_sub_pow] |
| rw [(PowerSeries.invOneSubPow ℤ m).inv_val, mul_one] |
|
|
| theorem regular_sequence_series_geometric [Fintype σ] |
| (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) = |
| ∏ i ∈ Finset.range rs.length, ∑ j ∈ Finset.range (degreeAt degrees i), |
| (PowerSeries.X : PowerSeries ℤ) ^ j := by |
| have h := regular_prefix_series rs degrees hhom hreg rs.length le_rfl |
| simp only [List.take_length] at h |
| rw [h, sectionHilbertSeries_bot hσ, ← hlen] |
| exact degree_factors_mul_invOneSubPow (degreeAt degrees) rs.length |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
|
|
