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:

• σ : Type* (indexing the variables)

• R : Type* [CommSemiring R] (the coefficients)

• s : σ →₀ ℕ, a function from σ to ℕ which is zero away from a finite set. This will give rise to a monomial in MvPolynomial σ R which mathematicians might call X^s

• a : R

• i : σ, with corresponding monomial X i, often denoted X_i by mathematicians

• p : MvPolynomial σ R

Tags

equivalence, isomorphism, morphism, ring hom, hom

@[simp]

theorem MvPolynomial.pUnitAlgEquiv_apply (R : Type u) [CommSemiring R] (p : MvPolynomial PUnit.{u_2 + 1} R) : ↑(MvPolynomial.pUnitAlgEquiv R) p = MvPolynomial.eval₂ Polynomial.C (fun x => Polynomial.X) p

@[simp]

theorem MvPolynomial.pUnitAlgEquiv_symm_apply (R : Type u) [CommSemiring R] (p : Polynomial R) : ↑(AlgEquiv.symm (MvPolynomial.pUnitAlgEquiv R)) p = Polynomial.eval₂ MvPolynomial.C (MvPolynomial.X PUnit.unit) p

def MvPolynomial.pUnitAlgEquiv (R : Type u) [CommSemiring R] : MvPolynomial PUnit.{u_2 + 1} R ≃ₐ[R] Polynomial R

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

@[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) : ↑(MvPolynomial.mapEquiv σ e) a = ↑(MvPolynomial.map ↑e) a

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

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

@[simp]

theorem MvPolynomial.mapEquiv_refl {R : Type u} (σ : Type u_1) [CommSemiring R] : MvPolynomial.mapEquiv σ (RingEquiv.refl R) = RingEquiv.refl (MvPolynomial σ R)

@[simp]

theorem MvPolynomial.mapEquiv_symm {S₁ : Type v} {S₂ : Type w} (σ : Type u_1) [CommSemiring S₁] [CommSemiring S₂] (e : S₁ ≃+* S₂) : RingEquiv.symm (MvPolynomial.mapEquiv σ e) = MvPolynomial.mapEquiv σ (RingEquiv.symm e)

@[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) : RingEquiv.trans (MvPolynomial.mapEquiv σ e) (MvPolynomial.mapEquiv σ f) = MvPolynomial.mapEquiv σ (RingEquiv.trans e f)

@[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) : ↑(MvPolynomial.mapAlgEquiv σ e) a = ↑(MvPolynomial.map ↑e) a

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) : MvPolynomial σ A₁ ≃ₐ[R] MvPolynomial σ A₂

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

@[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) : AlgEquiv.symm (MvPolynomial.mapAlgEquiv σ e) = MvPolynomial.mapAlgEquiv σ (AlgEquiv.symm e)

@[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) : AlgEquiv.trans (MvPolynomial.mapAlgEquiv σ e) (MvPolynomial.mapAlgEquiv σ f) = MvPolynomial.mapAlgEquiv σ (AlgEquiv.trans e f)

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.

@[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) : ↑(MvPolynomial.sumToIter R S₁ S₂) (MvPolynomial.X (Sum.inl b)) = MvPolynomial.X b

@[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.

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

theorem MvPolynomial.iterToSum_X (R : Type u) (S₁ : Type v) (S₂ : Type w) [CommSemiring R] (b : S₁) : ↑(MvPolynomial.iterToSum R S₁ S₂) (MvPolynomial.X b) = MvPolynomial.X (Sum.inl b)

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_toFun (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] (a : R) (j : σ →₀ ℕ) : ↑(↑(AlgEquiv.symm (MvPolynomial.isEmptyAlgEquiv R σ)) a) j = Pi.single 0 a j

@[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 => IsEmpty.elim he a) a

@[simp]

theorem MvPolynomial.isEmptyAlgEquiv_symm_apply_support (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] (a : R) : (↑(AlgEquiv.symm (MvPolynomial.isEmptyAlgEquiv R σ)) a).support = if a = 0 then ∅ else {0}

def MvPolynomial.isEmptyAlgEquiv (R : Type u) (σ : Type u_1) [CommSemiring R] [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R

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

@[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 => IsEmpty.elim inst✝ a) a

@[simp]

theorem MvPolynomial.isEmptyRingEquiv_symm_apply_support (R : Type u) (σ : Type u_1) [CommSemiring R] [IsEmpty σ] (a : R) : (↑(RingEquiv.symm (MvPolynomial.isEmptyRingEquiv R σ)) a).support = if a = 0 then ∅ else {0}

@[simp]

theorem MvPolynomial.isEmptyRingEquiv_symm_apply_toFun (R : Type u) (σ : Type u_1) [CommSemiring R] [IsEmpty σ] (a : R) (j : σ →₀ ℕ) : ↑(↑(RingEquiv.symm (MvPolynomial.isEmptyRingEquiv R σ)) a) j = Pi.single 0 a j

def MvPolynomial.isEmptyRingEquiv (R : Type u) (σ : Type u_1) [CommSemiring R] [IsEmpty σ] : MvPolynomial σ R ≃+* R

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

@[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 : RingHom.comp (RingHom.comp f g) MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), ↑f (↑g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : RingHom.comp (RingHom.comp g f) MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), ↑g (↑f (MvPolynomial.X n)) = MvPolynomial.X n) (a : MvPolynomial S₂ S₃) : ↑(RingEquiv.symm (MvPolynomial.mvPolynomialEquivMvPolynomial R S₁ S₂ S₃ f g hfgC hfgX hgfC hgfX)) 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 : RingHom.comp (RingHom.comp f g) MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), ↑f (↑g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : RingHom.comp (RingHom.comp g f) 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 : RingHom.comp (RingHom.comp f g) MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), ↑f (↑g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : RingHom.comp (RingHom.comp g f) MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), ↑g (↑f (MvPolynomial.X n)) = MvPolynomial.X n) : MvPolynomial S₁ R ≃+* MvPolynomial S₂ S₃

A helper function for sumRingEquiv.

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.

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.

@[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 => match o, Polynomial.X, fun s => ↑Polynomial.C (MvPolynomial.X s) with | some x, x_1, f => f x | none, y, x => y) a

@[simp]

theorem MvPolynomial.optionEquivLeft_symm_apply (R : Type u) (S₁ : Type v) [CommSemiring R] (a : Polynomial (MvPolynomial S₁ R)) : ↑(AlgEquiv.symm (MvPolynomial.optionEquivLeft R S₁)) a = ↑(Polynomial.aevalTower (MvPolynomial.rename some) (MvPolynomial.X none)) a

def MvPolynomial.optionEquivLeft (R : Type u) (S₁ : Type v) [CommSemiring R] : MvPolynomial (Option S₁) R ≃ₐ[R] Polynomial (MvPolynomial S₁ R)

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

def MvPolynomial.optionEquivRight (R : Type u) (S₁ : Type v) [CommSemiring R] : MvPolynomial (Option S₁) R ≃ₐ[R] MvPolynomial S₁ (Polynomial R)

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

def MvPolynomial.finSuccEquiv (R : Type u) [CommSemiring R] (n : ℕ) : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] Polynomial (MvPolynomial (Fin n) R)

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

theorem MvPolynomial.finSuccEquiv_eq (R : Type u) [CommSemiring R] (n : ℕ) : ↑(MvPolynomial.finSuccEquiv R n) = MvPolynomial.eval₂Hom (RingHom.comp Polynomial.C MvPolynomial.C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (MvPolynomial.X k)) i

@[simp]

theorem MvPolynomial.finSuccEquiv_apply (R : Type u) [CommSemiring R] (n : ℕ) (p : MvPolynomial (Fin (n + 1)) R) : ↑(MvPolynomial.finSuccEquiv R n) p = ↑(MvPolynomial.eval₂Hom (RingHom.comp Polynomial.C MvPolynomial.C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (MvPolynomial.X k)) i) p

theorem MvPolynomial.finSuccEquiv_comp_C_eq_C {R : Type u} [CommSemiring R] (n : ℕ) : RingHom.comp (↑(AlgEquiv.symm (MvPolynomial.finSuccEquiv R n))) (RingHom.comp Polynomial.C MvPolynomial.C) = MvPolynomial.C

theorem MvPolynomial.finSuccEquiv_X_zero {R : Type u} [CommSemiring R] {n : ℕ} : ↑(MvPolynomial.finSuccEquiv R n) (MvPolynomial.X 0) = Polynomial.X

theorem MvPolynomial.finSuccEquiv_X_succ {R : Type u} [CommSemiring R] {n : ℕ} {j : Fin n} : ↑(MvPolynomial.finSuccEquiv R n) (MvPolynomial.X (Fin.succ j)) = ↑Polynomial.C (MvPolynomial.X j)

theorem MvPolynomial.finSuccEquiv_coeff_coeff {R : Type u} [CommSemiring R] {n : ℕ} (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) : MvPolynomial.coeff m (Polynomial.coeff (↑(MvPolynomial.finSuccEquiv R n) f) i) = MvPolynomial.coeff (Finsupp.cons i m) f

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.eval_eq_eval_mv_eval' {R : Type u} [CommSemiring R] {n : ℕ} (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) : ↑(MvPolynomial.eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (MvPolynomial.eval s) (↑(MvPolynomial.finSuccEquiv R n) f))

theorem MvPolynomial.coeff_eval_eq_eval_coeff {R : Type u} [CommSemiring R] {n : ℕ} (s' : Fin n → R) (f : Polynomial (MvPolynomial (Fin n) R)) (i : ℕ) : Polynomial.coeff (Polynomial.map (MvPolynomial.eval s') f) i = ↑(MvPolynomial.eval s') (Polynomial.coeff f i)

theorem MvPolynomial.support_coeff_finSuccEquiv {R : Type u} [CommSemiring R] {n : ℕ} {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} : m ∈ MvPolynomial.support (Polynomial.coeff (↑(MvPolynomial.finSuccEquiv R n) f) i) ↔ Finsupp.cons i m ∈ MvPolynomial.support f

theorem MvPolynomial.finSuccEquiv_support {R : Type u} [CommSemiring R] {n : ℕ} (f : MvPolynomial (Fin (n + 1)) R) : Polynomial.support (↑(MvPolynomial.finSuccEquiv R n) f) = Finset.image (fun m => ↑m 0) (MvPolynomial.support f)

theorem MvPolynomial.finSuccEquiv_support' {R : Type u} [CommSemiring R] {n : ℕ} {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} : Finset.image (Finsupp.cons i) (MvPolynomial.support (Polynomial.coeff (↑(MvPolynomial.finSuccEquiv R n) f) i)) = Finset.filter (fun m => ↑m 0 = i) (MvPolynomial.support f)

theorem MvPolynomial.support_finSuccEquiv_nonempty {R : Type u} [CommSemiring R] {n : ℕ} {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) : Finset.Nonempty (Polynomial.support (↑(MvPolynomial.finSuccEquiv R n) f))

theorem MvPolynomial.degree_finSuccEquiv {R : Type u} [CommSemiring R] {n : ℕ} {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) : Polynomial.degree (↑(MvPolynomial.finSuccEquiv R n) f) = ↑(MvPolynomial.degreeOf 0 f)

theorem MvPolynomial.natDegree_finSuccEquiv {R : Type u} [CommSemiring R] {n : ℕ} (f : MvPolynomial (Fin (n + 1)) R) : Polynomial.natDegree (↑(MvPolynomial.finSuccEquiv R n) f) = MvPolynomial.degreeOf 0 f

theorem MvPolynomial.degreeOf_coeff_finSuccEquiv {R : Type u} [CommSemiring R] {n : ℕ} (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) : MvPolynomial.degreeOf j (Polynomial.coeff (↑(MvPolynomial.finSuccEquiv R n) p) i) ≤ MvPolynomial.degreeOf (Fin.succ j) p