Documentation

Mathlib.Algebra.MvPolynomial.Equiv

Equivalences between polynomial rings #

This file establishes a number of equivalences between polynomial rings, based on equivalences between the underlying types.

Notation #

As in other polynomial files, we typically use the notation:

Tags #

equivalence, isomorphism, morphism, ring hom, hom

The ring isomorphism between multivariable polynomials in a single variable and polynomials over the ground ring.

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    @[simp]
    theorem MvPolynomial.mapEquiv_apply {S₁ : Type v} {S₂ : Type w} (σ : Type u_1) [CommSemiring S₁] [CommSemiring S₂] (e : S₁ ≃+* S₂) (a : MvPolynomial σ S₁) :
    def MvPolynomial.mapEquiv {S₁ : Type v} {S₂ : Type w} (σ : Type u_1) [CommSemiring S₁] [CommSemiring S₂] (e : S₁ ≃+* S₂) :

    If e : A ≃+* B is an isomorphism of rings, then so is map e.

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      @[simp]
      theorem MvPolynomial.mapEquiv_symm {S₁ : Type v} {S₂ : Type w} (σ : Type u_1) [CommSemiring S₁] [CommSemiring S₂] (e : S₁ ≃+* S₂) :
      @[simp]
      theorem MvPolynomial.mapEquiv_trans {S₁ : Type v} {S₂ : Type w} {S₃ : Type x} (σ : Type u_1) [CommSemiring S₁] [CommSemiring S₂] [CommSemiring S₃] (e : S₁ ≃+* S₂) (f : S₂ ≃+* S₃) :
      @[simp]
      theorem MvPolynomial.mapAlgEquiv_apply {R : Type u} (σ : Type u_1) [CommSemiring R] {A₁ : Type u_2} {A₂ : Type u_3} [CommSemiring A₁] [CommSemiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) (a : MvPolynomial σ A₁) :
      def MvPolynomial.mapAlgEquiv {R : Type u} (σ : Type u_1) [CommSemiring R] {A₁ : Type u_2} {A₂ : Type u_3} [CommSemiring A₁] [CommSemiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) :

      If e : A ≃ₐ[R] B is an isomorphism of R-algebras, then so is map e.

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        @[simp]
        theorem MvPolynomial.mapAlgEquiv_refl {R : Type u} (σ : Type u_1) [CommSemiring R] {A₁ : Type u_2} [CommSemiring A₁] [Algebra R A₁] :
        MvPolynomial.mapAlgEquiv σ AlgEquiv.refl = AlgEquiv.refl
        @[simp]
        theorem MvPolynomial.mapAlgEquiv_symm {R : Type u} (σ : Type u_1) [CommSemiring R] {A₁ : Type u_2} {A₂ : Type u_3} [CommSemiring A₁] [CommSemiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) :
        @[simp]
        theorem MvPolynomial.mapAlgEquiv_trans {R : Type u} (σ : Type u_1) [CommSemiring R] {A₁ : Type u_2} {A₂ : Type u_3} {A₃ : Type u_4} [CommSemiring A₁] [CommSemiring A₂] [CommSemiring A₃] [Algebra R A₁] [Algebra R A₂] [Algebra R A₃] (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
        def MvPolynomial.sumToIter (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] :
        MvPolynomial (S₁ S₂) R →+* MvPolynomial S₁ (MvPolynomial S₂ R)

        The function from multivariable polynomials in a sum of two types, to multivariable polynomials in one of the types, with coefficients in multivariable polynomials in the other type.

        See sumRingEquiv for the ring isomorphism.

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          @[simp]
          theorem MvPolynomial.sumToIter_C (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (a : R) :
          (MvPolynomial.sumToIter R S₁ S₂) (MvPolynomial.C a) = MvPolynomial.C (MvPolynomial.C a)
          @[simp]
          theorem MvPolynomial.sumToIter_Xl (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (b : S₁) :
          @[simp]
          theorem MvPolynomial.sumToIter_Xr (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (c : S₂) :
          (MvPolynomial.sumToIter R S₁ S₂) (MvPolynomial.X (Sum.inr c)) = MvPolynomial.C (MvPolynomial.X c)
          def MvPolynomial.iterToSum (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] :
          MvPolynomial S₁ (MvPolynomial S₂ R) →+* MvPolynomial (S₁ S₂) R

          The function from multivariable polynomials in one type, with coefficients in multivariable polynomials in another type, to multivariable polynomials in the sum of the two types.

          See sumRingEquiv for the ring isomorphism.

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            @[simp]
            theorem MvPolynomial.iterToSum_C_C (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (a : R) :
            (MvPolynomial.iterToSum R S₁ S₂) (MvPolynomial.C (MvPolynomial.C a)) = MvPolynomial.C a
            @[simp]
            theorem MvPolynomial.iterToSum_X (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (b : S₁) :
            @[simp]
            theorem MvPolynomial.iterToSum_C_X (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (c : S₂) :
            (MvPolynomial.iterToSum R S₁ S₂) (MvPolynomial.C (MvPolynomial.X c)) = MvPolynomial.X (Sum.inr c)
            @[simp]
            theorem MvPolynomial.isEmptyAlgEquiv_symm_apply_support (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] (a : R) :
            ((MvPolynomial.isEmptyAlgEquiv R σ).symm a).support = if a = 0 then else {0}
            @[simp]
            theorem MvPolynomial.isEmptyAlgEquiv_apply (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] (a : MvPolynomial σ R) :
            (MvPolynomial.isEmptyAlgEquiv R σ) a = (MvPolynomial.aeval fun (a : σ) => he.elim a) a
            @[simp]
            theorem MvPolynomial.isEmptyAlgEquiv_symm_apply_toFun (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] (a : R) (j : σ →₀ ) :
            ((MvPolynomial.isEmptyAlgEquiv R σ).symm a) j = Pi.single 0 a j
            def MvPolynomial.isEmptyAlgEquiv (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] :

            The algebra isomorphism between multivariable polynomials in no variables and the ground ring.

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              @[simp]
              theorem MvPolynomial.isEmptyRingEquiv_symm_apply_toFun (R : Type u) (σ : Type u_1) [CommSemiring R] [IsEmpty σ] (a : R) (j : σ →₀ ) :
              ((MvPolynomial.isEmptyRingEquiv R σ).symm a) j = Pi.single 0 a j
              @[simp]
              theorem MvPolynomial.isEmptyRingEquiv_symm_apply_support (R : Type u) (σ : Type u_1) [CommSemiring R] [IsEmpty σ] (a : R) :
              ((MvPolynomial.isEmptyRingEquiv R σ).symm a).support = if a = 0 then else {0}
              @[simp]
              theorem MvPolynomial.isEmptyRingEquiv_apply (R : Type u) (σ : Type u_1) [CommSemiring R] [IsEmpty σ] (a : MvPolynomial σ R) :
              (MvPolynomial.isEmptyRingEquiv R σ) a = (MvPolynomial.aeval fun (a : σ) => inst✝.elim a) a

              The ring isomorphism between multivariable polynomials in no variables and the ground ring.

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                @[simp]
                theorem MvPolynomial.mvPolynomialEquivMvPolynomial_symm_apply (R : Type u) (S₁ : Type v) (S₂ : Type w) (S₃ : Type x) [CommSemiring R] [CommSemiring S₃] (f : MvPolynomial S₁ R →+* MvPolynomial S₂ S₃) (g : MvPolynomial S₂ S₃ →+* MvPolynomial S₁ R) (hfgC : (f.comp g).comp MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), f (g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : (g.comp f).comp MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), g (f (MvPolynomial.X n)) = MvPolynomial.X n) (a : MvPolynomial S₂ S₃) :
                (MvPolynomial.mvPolynomialEquivMvPolynomial R S₁ S₂ S₃ f g hfgC hfgX hgfC hgfX).symm a = g a
                @[simp]
                theorem MvPolynomial.mvPolynomialEquivMvPolynomial_apply (R : Type u) (S₁ : Type v) (S₂ : Type w) (S₃ : Type x) [CommSemiring R] [CommSemiring S₃] (f : MvPolynomial S₁ R →+* MvPolynomial S₂ S₃) (g : MvPolynomial S₂ S₃ →+* MvPolynomial S₁ R) (hfgC : (f.comp g).comp MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), f (g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : (g.comp f).comp MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), g (f (MvPolynomial.X n)) = MvPolynomial.X n) (a : MvPolynomial S₁ R) :
                (MvPolynomial.mvPolynomialEquivMvPolynomial R S₁ S₂ S₃ f g hfgC hfgX hgfC hgfX) a = f a
                def MvPolynomial.mvPolynomialEquivMvPolynomial (R : Type u) (S₁ : Type v) (S₂ : Type w) (S₃ : Type x) [CommSemiring R] [CommSemiring S₃] (f : MvPolynomial S₁ R →+* MvPolynomial S₂ S₃) (g : MvPolynomial S₂ S₃ →+* MvPolynomial S₁ R) (hfgC : (f.comp g).comp MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), f (g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : (g.comp f).comp MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), g (f (MvPolynomial.X n)) = MvPolynomial.X n) :

                A helper function for sumRingEquiv.

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                  def MvPolynomial.sumRingEquiv (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] :
                  MvPolynomial (S₁ S₂) R ≃+* MvPolynomial S₁ (MvPolynomial S₂ R)

                  The ring isomorphism between multivariable polynomials in a sum of two types, and multivariable polynomials in one of the types, with coefficients in multivariable polynomials in the other type.

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                    @[simp]
                    theorem MvPolynomial.sumAlgEquiv_apply (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (a : MvPolynomial (S₁ S₂) R) :
                    (MvPolynomial.sumAlgEquiv R S₁ S₂) a = (MvPolynomial.sumToIter R S₁ S₂) a
                    @[simp]
                    theorem MvPolynomial.sumAlgEquiv_symm_apply (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (a : MvPolynomial S₁ (MvPolynomial S₂ R)) :
                    (MvPolynomial.sumAlgEquiv R S₁ S₂).symm a = (MvPolynomial.iterToSum R S₁ S₂) a
                    def MvPolynomial.sumAlgEquiv (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] :
                    MvPolynomial (S₁ S₂) R ≃ₐ[R] MvPolynomial S₁ (MvPolynomial S₂ R)

                    The algebra isomorphism between multivariable polynomials in a sum of two types, and multivariable polynomials in one of the types, with coefficients in multivariable polynomials in the other type.

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                      @[simp]
                      theorem MvPolynomial.optionEquivLeft_apply (R : Type u) (S₁ : Type v) [CommSemiring R] (a : MvPolynomial (Option S₁) R) :
                      (MvPolynomial.optionEquivLeft R S₁) a = (MvPolynomial.aeval fun (o : Option S₁) => o.elim Polynomial.X fun (s : S₁) => Polynomial.C (MvPolynomial.X s)) a

                      The algebra isomorphism between multivariable polynomials in Option S₁ and polynomials with coefficients in MvPolynomial S₁ R.

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                        theorem MvPolynomial.optionEquivLeft_X_some (R : Type u) (S₁ : Type v) [CommSemiring R] (x : S₁) :
                        theorem MvPolynomial.optionEquivLeft_C (R : Type u) (S₁ : Type v) [CommSemiring R] (r : R) :
                        (MvPolynomial.optionEquivLeft R S₁) (MvPolynomial.C r) = Polynomial.C (MvPolynomial.C r)
                        @[simp]
                        theorem MvPolynomial.optionEquivRight_apply (R : Type u) (S₁ : Type v) [CommSemiring R] (a : MvPolynomial (Option S₁) R) :
                        (MvPolynomial.optionEquivRight R S₁) a = (MvPolynomial.aeval fun (o : Option S₁) => o.elim (MvPolynomial.C Polynomial.X) MvPolynomial.X) a

                        The algebra isomorphism between multivariable polynomials in Option S₁ and multivariable polynomials with coefficients in polynomials.

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                          theorem MvPolynomial.optionEquivRight_X_none (R : Type u) (S₁ : Type v) [CommSemiring R] :
                          (MvPolynomial.optionEquivRight R S₁) (MvPolynomial.X none) = MvPolynomial.C Polynomial.X
                          theorem MvPolynomial.optionEquivRight_C (R : Type u) (S₁ : Type v) [CommSemiring R] (r : R) :
                          (MvPolynomial.optionEquivRight R S₁) (MvPolynomial.C r) = MvPolynomial.C (Polynomial.C r)

                          The algebra isomorphism between multivariable polynomials in Fin (n + 1) and polynomials over multivariable polynomials in Fin n.

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                            theorem MvPolynomial.finSuccEquiv_eq (R : Type u) [CommSemiring R] (n : ) :
                            (MvPolynomial.finSuccEquiv R n) = MvPolynomial.eval₂Hom (Polynomial.C.comp MvPolynomial.C) fun (i : Fin (n + 1)) => Fin.cases Polynomial.X (fun (k : Fin n) => Polynomial.C (MvPolynomial.X k)) i
                            @[simp]
                            theorem MvPolynomial.finSuccEquiv_apply (R : Type u) [CommSemiring R] (n : ) (p : MvPolynomial (Fin (n + 1)) R) :
                            (MvPolynomial.finSuccEquiv R n) p = (MvPolynomial.eval₂Hom (Polynomial.C.comp MvPolynomial.C) fun (i : Fin (n + 1)) => Fin.cases Polynomial.X (fun (k : Fin n) => Polynomial.C (MvPolynomial.X k)) i) p
                            theorem MvPolynomial.finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ) :
                            ((MvPolynomial.finSuccEquiv R n).symm).comp (Polynomial.C.comp MvPolynomial.C) = MvPolynomial.C

                            The coefficient of m in the i-th coefficient of finSuccEquiv R n f equals the coefficient of Finsupp.cons i m in f.

                            theorem MvPolynomial.coeff_eval_eq_eval_coeff {R : Type u} [CommSemiring R] {n : } (s' : Fin nR) (f : Polynomial (MvPolynomial (Fin n) R)) (i : ) :
                            (Polynomial.map (MvPolynomial.eval s') f).coeff i = (MvPolynomial.eval s') (f.coeff i)
                            theorem MvPolynomial.support_coeff_finSuccEquiv {R : Type u} [CommSemiring R] {n : } {f : MvPolynomial (Fin (n + 1)) R} {i : } {m : Fin n →₀ } :
                            m (((MvPolynomial.finSuccEquiv R n) f).coeff i).support Finsupp.cons i m f.support
                            theorem MvPolynomial.totalDegree_coeff_finSuccEquiv_add_le {R : Type u} [CommSemiring R] {n : } (f : MvPolynomial (Fin (n + 1)) R) (i : ) (hi : ((MvPolynomial.finSuccEquiv R n) f).coeff i 0) :
                            (((MvPolynomial.finSuccEquiv R n) f).coeff i).totalDegree + i f.totalDegree

                            The totalDegree of a multivariable polynomial p is at least i more than the totalDegree of the ith coefficient of finSuccEquiv applied to p, if this is nonzero.

                            theorem MvPolynomial.finSuccEquiv_support {R : Type u} [CommSemiring R] {n : } (f : MvPolynomial (Fin (n + 1)) R) :
                            ((MvPolynomial.finSuccEquiv R n) f).support = Finset.image (fun (m : Fin (n + 1) →₀ ) => m 0) f.support
                            theorem MvPolynomial.finSuccEquiv_support' {R : Type u} [CommSemiring R] {n : } {f : MvPolynomial (Fin (n + 1)) R} {i : } :
                            Finset.image (Finsupp.cons i) (((MvPolynomial.finSuccEquiv R n) f).coeff i).support = Finset.filter (fun (m : Fin (n + 1) →₀ ) => m 0 = i) f.support
                            theorem MvPolynomial.support_finSuccEquiv_nonempty {R : Type u} [CommSemiring R] {n : } {f : MvPolynomial (Fin (n + 1)) R} (h : f 0) :
                            ((MvPolynomial.finSuccEquiv R n) f).support.Nonempty
                            theorem MvPolynomial.degree_finSuccEquiv {R : Type u} [CommSemiring R] {n : } {f : MvPolynomial (Fin (n + 1)) R} (h : f 0) :
                            theorem MvPolynomial.finSuccEquiv_rename_finSuccEquiv {R : Type u} {σ : Type u_1} [CommSemiring R] {n : } (e : σ Fin n) (φ : MvPolynomial (Option σ) R) :
                            (MvPolynomial.finSuccEquiv R n) ((MvPolynomial.rename (e.optionCongr.trans (finSuccEquiv n).symm)) φ) = Polynomial.map (MvPolynomial.rename e).toRingHom ((MvPolynomial.optionEquivLeft R σ) φ)

                            Consider a multivariate polynomial φ whose variables are indexed by Option σ, and suppose that σ ≃ Fin n. Then one may view φ as a polynomial over MvPolynomial (Fin n) R, by

                            1. renaming the variables via Option σ ≃ Fin (n+1), and then singling out the 0-th variable via MvPolynomial.finSuccEquiv;
                            2. first viewing it as polynomial over MvPolynomial σ R via MvPolynomial.optionEquivLeft, and then renaming the variables.

                            This lemma shows that both constructions are the same.