| import OAI.NumberTheory.PiExponent.Polynomials.PowerCoverSubstitution |
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| namespace OAI |
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| namespace PiExponent.PowerCover |
| noncomputable section |
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| variable {σ : Type*} [Fintype σ] {k : Type*} [CommSemiring k] |
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| def decompositionAddEquiv (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| MvPolynomial σ k ≃+ ((∀ i, Fin (w i)) →₀ MvPolynomial σ k) := |
| (AddMonoidAlgebra.coeffAddEquiv : MvPolynomial σ k ≃+ ((σ →₀ ℕ) →₀ k)).trans <| |
| (Finsupp.domCongr ((exponentEquiv w hw).trans (Equiv.prodComm _ _))).trans <| |
| Finsupp.curryAddEquiv.trans <| |
| Finsupp.mapRange.addEquiv AddMonoidAlgebra.coeffAddEquiv.symm |
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| @[simp] theorem decompositionAddEquiv_monomial (w : σ → ℕ) (hw : ∀ i, 0 < w i) |
| (a : σ →₀ ℕ) (c : k) : |
| decompositionAddEquiv w hw (MvPolynomial.monomial a c) = |
| Finsupp.single (residue w hw a) (MvPolynomial.monomial (quotient w a) c) := by |
| classical |
| simp [decompositionAddEquiv, MvPolynomial.monomial, Finsupp.curry_single] |
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| theorem decompositionAddEquiv_power_mul (w : σ → ℕ) (hw : ∀ i, 0 < w i) |
| (p q : MvPolynomial σ k) : |
| decompositionAddEquiv w hw (powerSubstitution w p * q) = |
| p • decompositionAddEquiv w hw q := by |
| classical |
| induction p using MvPolynomial.induction_on' with |
| | monomial a c => |
| induction q using MvPolynomial.induction_on' with |
| | monomial b d => |
| simp only [powerSubstitution_monomial, MvPolynomial.monomial_mul_monomial, |
| decompositionAddEquiv_monomial] |
| rw [add_comm (scale w a) b, quotient_add_scale w hw, residue_add_scale] |
| simp [Finsupp.smul_single, smul_eq_mul, MvPolynomial.monomial_mul_monomial, add_comm] |
| | add q r hq hr => |
| simp only [mul_add, map_add, smul_add, hq, hr] |
| | add p r hp hr => |
| simp only [map_add, add_mul, add_smul, hp, hr] |
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| def PowerModule (_weights : σ → ℕ) (k : Type*) [CommSemiring k] := MvPolynomial σ k |
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| instance (w : σ → ℕ) : CommSemiring (PowerModule w k) := |
| inferInstanceAs (CommSemiring (MvPolynomial σ k)) |
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| def toPolynomial (w : σ → ℕ) : PowerModule w k ≃+* MvPolynomial σ k := |
| RingEquiv.refl _ |
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| instance powerAlgebra (w : σ → ℕ) : Algebra (MvPolynomial σ k) (PowerModule w k) := |
| ((toPolynomial w).symm.toRingHom.comp |
| (powerSubstitution w).toRingHom).toAlgebra |
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| omit [Fintype σ] in |
| @[simp] theorem toPolynomial_smul (w : σ → ℕ) (p : MvPolynomial σ k) |
| (q : PowerModule w k) : |
| toPolynomial w (p • q) = |
| powerSubstitution w p * toPolynomial w q := rfl |
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| def decomposition (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| PowerModule w k ≃ₗ[MvPolynomial σ k] ((∀ i, Fin (w i)) →₀ MvPolynomial σ k) := |
| ((toPolynomial w).toAddEquiv.trans (decompositionAddEquiv w hw)).toLinearEquiv <| by |
| intro p q |
| exact decompositionAddEquiv_power_mul w hw p (toPolynomial w q) |
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| def basis (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| Module.Basis (∀ i, Fin (w i)) (MvPolynomial σ k) (PowerModule w k) where |
| repr := decomposition w hw |
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| theorem basis_toPolynomial (w : σ → ℕ) (hw : ∀ i, 0 < w i) |
| (r : ∀ i, Fin (w i)) : |
| toPolynomial w (basis (k := k) w hw r) = MvPolynomial.monomial (combine w (0, r)) 1 := by |
| apply (decompositionAddEquiv w hw).injective |
| rw [decompositionAddEquiv_monomial, quotient_combine w hw, residue_combine] |
| change (basis (k := k) w hw).repr (basis w hw r) = _ |
| simp |
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| theorem module_free (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| Module.Free (MvPolynomial σ k) (PowerModule w k) := |
| Module.Free.of_basis (basis w hw) |
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| theorem module_finite (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| Module.Finite (MvPolynomial σ k) (PowerModule w k) := by |
| classical |
| exact Module.Finite.of_basis (basis w hw) |
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| theorem finrank (w : σ → ℕ) (hw : ∀ i, 0 < w i) |
| [StrongRankCondition (MvPolynomial σ k)] : |
| Module.finrank (MvPolynomial σ k) (PowerModule w k) = ∏ i, w i := by |
| classical |
| rw [Module.finrank_eq_card_basis (basis w hw)] |
| simp [Fintype.card_pi] |
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| end |
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| noncomputable instance {σ k : Type*} [CommRing k] (w : σ → ℕ) : |
| CommRing (PowerModule w k) := |
| inferInstanceAs (CommRing (MvPolynomial σ k)) |
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| end PiExponent.PowerCover |
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| end OAI |
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