| import Mathlib.RingTheory.PowerSeries.Basic |
| import Mathlib.Tactic.NormNum |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularHilbertStep |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| variable {k σ : Type*} [Field k] [Finite σ] |
|
|
| noncomputable def sectionHilbertSeries (I : Ideal (MvPolynomial σ k)) : PowerSeries ℤ := |
| PowerSeries.mk (fun n => (Module.finrank k (quotientSection I n) : ℤ)) |
|
|
| omit [Finite σ] in |
| @[simp] theorem coeff_sectionHilbertSeries (I : Ideal (MvPolynomial σ k)) (n : ℕ) : |
| PowerSeries.coeff n (sectionHilbertSeries I) = |
| (Module.finrank k (quotientSection I n) : ℤ) := |
| PowerSeries.coeff_mk _ _ |
|
|
| theorem sectionHilbertSeries_regular_step (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) |
| (hreg : IsRightRegular (Ideal.Quotient.mk I f)) : |
| sectionHilbertSeries (Ideal.span {f} ⊔ I) = |
| (1 - PowerSeries.X ^ d) * sectionHilbertSeries I := by |
| ext n |
| rw [sub_mul, one_mul, map_sub, PowerSeries.coeff_X_pow_mul'] |
| simp only [coeff_sectionHilbertSeries] |
| by_cases hdn : d ≤ n |
| · rw [ite_eq_left hdn] |
| have h := regular_hilbert_step I hI f hf hreg (n-d) |
| rw [Nat.sub_add_cancel hdn] at h |
| apply eq_sub_of_add_eq |
| exact_mod_cast h |
| · rw [ite_eq_right hdn, sub_zero] |
| exact congrArg (fun m : ℕ => (m : ℤ)) |
| (hilbert_step_of_degree_lt I hI f hf (Nat.lt_of_not_ge hdn)) |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
|
|
