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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 | 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
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