| import Lean.Elab.Tactic.Omega |
| import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas |
| import Mathlib.RingTheory.Ideal.Colon |
| import OAI.NumberTheory.PiExponent.Analysis.HilbertSeries |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| attribute [local instance] MvPolynomial.gradedAlgebra |
| variable {k σ : Type*} [Field k] [Finite σ] |
|
|
| omit [Finite σ] in |
| theorem ideal_le_colon_singleton (I : Ideal (MvPolynomial σ k)) |
| (f : MvPolynomial σ k) : I ≤ I.colon {f} := by |
| intro p hp |
| apply Submodule.mem_colon_singleton.mpr |
| simpa only [smul_eq_mul] using I.mul_mem_right f hp |
|
|
| omit [Finite σ] in |
| theorem quotientSectionFactor_colon_ker (I : Ideal (MvPolynomial σ k)) |
| {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) (n : ℕ) : |
| LinearMap.ker (quotientSectionFactor I (I.colon {f}) |
| (ideal_le_colon_singleton I f) n) = |
| LinearMap.ker (quotientSectionMultiply I f hf n) := by |
| ext x |
| obtain ⟨p, hp, hpx⟩ := Submodule.mem_map.mp x.2 |
| change Ideal.Quotient.mk I p = x.1 at hpx |
| simp only [LinearMap.mem_ker, Subtype.ext_iff] |
| change (Ideal.Quotient.factorₐ k (ideal_le_colon_singleton I f)) x.1 = 0 ↔ |
| x.1 * Ideal.Quotient.mk I f = 0 |
| rw [← hpx] |
| change Ideal.Quotient.mk (I.colon {f}) p = 0 ↔ |
| Ideal.Quotient.mk I (p * f) = 0 |
| rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.Quotient.eq_zero_iff_mem, |
| Submodule.mem_colon_singleton, smul_eq_mul] |
|
|
| theorem quotientSectionMultiply_range_finrank_colon |
| (I : Ideal (MvPolynomial σ k)) {d : ℕ} |
| (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) (n : ℕ) : |
| Module.finrank k (LinearMap.range (quotientSectionMultiply I f hf n)) = |
| Module.finrank k (quotientSection (I.colon {f}) n) := by |
| let φ := quotientSectionFactor I (I.colon {f}) (ideal_le_colon_singleton I f) n |
| have hd := (LinearMap.ker φ).finrank_quotient_add_finrank |
| let e := φ.quotKerEquivOfSurjective |
| (quotientSectionFactor_surjective I (I.colon {f}) (ideal_le_colon_singleton I f) n) |
| rw [e.finrank_eq] at hd |
| change Module.finrank k (quotientSection (I.colon {f}) n) + |
| Module.finrank k (LinearMap.ker (quotientSectionFactor I (I.colon {f}) |
| (ideal_le_colon_singleton I f) n)) = _ at hd |
| rw [quotientSectionFactor_colon_ker I f hf n] at hd |
| have hm := (quotientSectionMultiply I f hf n).finrank_range_add_finrank_ker |
| omega |
|
|
| theorem colon_hilbert_step (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) (n : ℕ) : |
| Module.finrank k (quotientSection (Ideal.span {f} ⊔ I) (n+d)) + |
| Module.finrank k (quotientSection (I.colon {f}) n) = |
| Module.finrank k (quotientSection I (n+d)) := by |
| have hd := (LinearMap.range (quotientSectionMultiply I f hf n)).finrank_quotient_add_finrank |
| rw [quotientSectionMultiply_range_finrank_colon I f hf n, |
| ← quotientSectionFactor_ker_eq I hI f hf n] at hd |
| let e := (quotientSectionFactor I (Ideal.span {f} ⊔ I) le_sup_right (n+d)).quotKerEquivOfSurjective |
| (quotientSectionFactor_surjective I (Ideal.span {f} ⊔ I) le_sup_right (n+d)) |
| rw [e.finrank_eq] at hd |
| exact hd |
|
|
| theorem sectionHilbertSeries_colon_step (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) : |
| sectionHilbertSeries I = sectionHilbertSeries (Ideal.span {f} ⊔ I) + |
| PowerSeries.X ^ d * sectionHilbertSeries (I.colon {f}) := by |
| ext n |
| rw [map_add, PowerSeries.coeff_X_pow_mul'] |
| simp only [coeff_sectionHilbertSeries] |
| by_cases hdn : d ≤ n |
| · rw [ite_eq_left hdn] |
| have h := colon_hilbert_step I hI f hf (n-d) |
| rw [Nat.sub_add_cancel hdn] at h |
| exact_mod_cast h.symm |
| · rw [ite_eq_right hdn, add_zero] |
| exact congrArg (fun m : ℕ => (m : ℤ)) |
| (hilbert_step_of_degree_lt I hI f hf (Nat.lt_of_not_ge hdn)).symm |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
|
|
