| import OAI.NumberTheory.PiExponent.Polynomials.PowerCoverExponents |
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| namespace OAI |
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| namespace PiExponent.PowerCover |
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| noncomputable section |
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| variable {σ k : Type*} [Fintype σ] [CommSemiring k] |
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| def scaleHom (w : σ → ℕ) : (σ →₀ ℕ) →+ (σ →₀ ℕ) where |
| toFun := scale w |
| map_zero' := scale_zero w |
| map_add' := scale_add w |
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| @[simp] theorem scaleHom_apply (w : σ → ℕ) (a : σ →₀ ℕ) : |
| scaleHom w a = scale w a := rfl |
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| @[simp] theorem scale_single (w : σ → ℕ) (i : σ) (n : ℕ) : |
| scale w (Finsupp.single i n) = Finsupp.single i (w i * n) := by |
| classical |
| ext j |
| by_cases h : i = j |
| · subst j; simp |
| · simp [h] |
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| theorem scale_injective (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| Function.Injective (scale w) := by |
| intro a b hab |
| ext i |
| have h := congrArg (fun c : σ →₀ ℕ => c i) hab |
| simp only [scale_apply] at h |
| exact Nat.eq_of_mul_eq_mul_left (hw i) h |
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| def powerSubstitution (w : σ → ℕ) : |
| MvPolynomial σ k →ₐ[k] MvPolynomial σ k := |
| MvPolynomial.aeval (fun i => MvPolynomial.X i ^ w i) |
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| omit [Fintype σ] in |
| @[simp] theorem powerSubstitution_X (w : σ → ℕ) (i : σ) : |
| powerSubstitution (k := k) w (MvPolynomial.X i) = MvPolynomial.X i ^ w i := by |
| simp [powerSubstitution] |
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| theorem powerSubstitution_eq_mapDomainAlgHom (w : σ → ℕ) : |
| powerSubstitution (k := k) w = AddMonoidAlgebra.mapDomainAlgHom k k (scaleHom w) := by |
| apply MvPolynomial.algHom_ext |
| intro i |
| rw [powerSubstitution_X, MvPolynomial.X_pow_eq_monomial] |
| change MvPolynomial.monomial (Finsupp.single i (w i)) (1 : k) = |
| AddMonoidAlgebra.mapDomain (scaleHom w) (AddMonoidAlgebra.single (Finsupp.single i 1) 1) |
| rw [AddMonoidAlgebra.mapDomain_single] |
| simp [MvPolynomial.single_eq_monomial] |
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| @[simp] theorem powerSubstitution_monomial (w : σ → ℕ) (a : σ →₀ ℕ) (c : k) : |
| powerSubstitution w (MvPolynomial.monomial a c) = |
| MvPolynomial.monomial (scale w a) c := by |
| rw [powerSubstitution_eq_mapDomainAlgHom] |
| change AddMonoidAlgebra.mapDomain (scaleHom w) (AddMonoidAlgebra.single a c) = _ |
| rw [AddMonoidAlgebra.mapDomain_single] |
| rfl |
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| theorem powerSubstitution_injective (w : σ → ℕ) (hw : ∀ i, 0 < w i) : |
| Function.Injective (powerSubstitution (k := k) w) := by |
| rw [powerSubstitution_eq_mapDomainAlgHom] |
| exact AddMonoidAlgebra.mapDomain_injective (scale_injective w hw) |
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| end |
| end PiExponent.PowerCover |
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| end OAI |
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