| import Lean.Elab.Tactic.Omega |
| import Mathlib.Data.Finsupp.Multiset |
| import Mathlib.Data.Sym.Card |
| import Mathlib.RingTheory.PowerSeries.WellKnown |
| import OAI.NumberTheory.PiExponent.Analysis.HilbertSeries |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| variable {σ : Type*} |
| |
| noncomputable def exponentDegreeEquivSym (n : ℕ) : |
| {d : σ →₀ ℕ | d.degree = n} ≃ Sym σ n := by |
| classical |
| exact |
| { toFun := fun d => ⟨d.1.toMultiset, by |
| rw [Finsupp.card_toMultiset] |
| exact d.2⟩ |
| invFun := fun m => ⟨Multiset.toFinsupp m.1, by |
| have hc := Finsupp.card_toMultiset (Multiset.toFinsupp m.1) |
| rw [Multiset.toFinsupp_toMultiset] at hc |
| change (Multiset.toFinsupp m.1).degree = n |
| exact hc.symm.trans m.2⟩ |
| left_inv := fun d => Subtype.ext (Multiset.toFinsupp.apply_symm_apply d.1) |
| right_inv := fun m => Subtype.ext (Multiset.toFinsupp.symm_apply_apply m.1) } |
|
|
| theorem card_exponents_degree [Fintype σ] (n : ℕ) : |
| Nat.card {d : σ →₀ ℕ | d.degree = n} = (Fintype.card σ).multichoose n := by |
| classical |
| rw [Nat.card_congr (exponentDegreeEquivSym n), Nat.card_eq_fintype_card] |
| exact Sym.card_sym_eq_multichoose σ n |
|
|
| variable {k : Type*} [Field k] [Finite σ] |
|
|
| omit [Finite σ] in |
| theorem quotientSection_bot_finrank (n : ℕ) : |
| Module.finrank k (quotientSection (⊥ : Ideal (MvPolynomial σ k)) n) = |
| Module.finrank k (MvPolynomial.homogeneousSubmodule σ k n) := by |
| let e := LinearEquiv.ofBijective |
| (Ideal.Quotient.mkₐ k (⊥ : Ideal (MvPolynomial σ k))).toLinearMap |
| ((Ideal.Quotient.mk_bijective_iff_eq_bot (⊥ : Ideal (MvPolynomial σ k))).mpr rfl) |
| exact e.finrank_map_eq (MvPolynomial.homogeneousSubmodule σ k n) |
|
|
| theorem quotientSection_bot_finrank_multichoose [Fintype σ] (n : ℕ) : |
| Module.finrank k (quotientSection (⊥ : Ideal (MvPolynomial σ k)) n) = |
| (Fintype.card σ).multichoose n := by |
| rw [quotientSection_bot_finrank, homogeneousSection_finrank, card_exponents_degree] |
|
|
| theorem sectionHilbertSeries_bot [Fintype σ] (hσ : 0 < Fintype.card σ) : |
| sectionHilbertSeries (⊥ : Ideal (MvPolynomial σ k)) = |
| (PowerSeries.invOneSubPow ℤ (Fintype.card σ)).val := by |
| rw [PowerSeries.invOneSubPow_val_eq_mk_sub_one_add_choose_of_pos ℤ (Fintype.card σ) hσ] |
| ext n |
| rw [coeff_sectionHilbertSeries, PowerSeries.coeff_mk, |
| quotientSection_bot_finrank_multichoose, Nat.multichoose_eq] |
| congr 1 |
| have hs : Fintype.card σ + n - 1 = (Fintype.card σ - 1) + n := by omega |
| rw [hs] |
| exact Nat.choose_symm_add.symm |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
|
|
