Expand multivariate polynomials

Given a multivariate polynomial φ, one may replace every occurrence of X i by X i ^ n, for some natural number n. This operation is called MvPolynomial.expand and it is an algebra homomorphism.

Main declaration

• MvPolynomial.expand: expand a polynomial by a factor of p, so ∑ aₙ xⁿ becomes ∑ aₙ xⁿᵖ.

noncomputable def MvPolynomial.expand {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolynomial σ R

Expand the polynomial by a factor of p, so ∑ aₙ xⁿ becomes ∑ aₙ xⁿᵖ.

See also Polynomial.expand.

Equations

• MvPolynomial.expand p = MvPolynomial.bind₁ fun (i : σ) => MvPolynomial.X i ^ p

theorem MvPolynomial.expand_C {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p : ℕ) (r : R) : (expand p) (C r) = C r

@[simp]

theorem MvPolynomial.expand_X {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p : ℕ) (i : σ) : (expand p) (X i) = X i ^ p

@[simp]

theorem MvPolynomial.expand_monomial {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p : ℕ) (d : σ →₀ ℕ) (r : R) : (expand p) ((monomial d) r) = (monomial (p • d)) r

@[simp]

theorem MvPolynomial.expand_zero {σ : Type u_1} {R : Type u_3} [CommSemiring R] : expand 0 = (Algebra.ofId R (MvPolynomial σ R)).comp (aeval 1)

theorem MvPolynomial.expand_zero_apply {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p : MvPolynomial σ R) : (expand 0) p = C ((eval 1) p)

@[simp]

theorem MvPolynomial.expand_one {σ : Type u_1} {R : Type u_3} [CommSemiring R] : expand 1 = AlgHom.id R (MvPolynomial σ R)

theorem MvPolynomial.expand_one_apply {σ : Type u_1} {R : Type u_3} [CommSemiring R] (f : MvPolynomial σ R) : (expand 1) f = f

theorem MvPolynomial.expand_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (p : ℕ) (f : σ → MvPolynomial τ R) : (expand p).comp (bind₁ f) = bind₁ fun (i : σ) => (expand p) (f i)

theorem MvPolynomial.expand_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (p : ℕ) (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : (expand p) ((bind₁ f) φ) = (bind₁ fun (i : σ) => (expand p) (f i)) φ

@[simp]

theorem MvPolynomial.map_expand {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (p : ℕ) (φ : MvPolynomial σ R) : (map f) ((expand p) φ) = (expand p) ((map f) φ)

@[simp]

theorem MvPolynomial.rename_comp_expand {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → τ) (p : ℕ) : (rename f).comp (expand p) = (expand p).comp (rename f)

@[simp]

theorem MvPolynomial.rename_expand {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [CommSemiring R] (f : σ → τ) (p : ℕ) (φ : MvPolynomial σ R) : (rename f) ((expand p) φ) = (expand p) ((rename f) φ)

theorem MvPolynomial.eval₂Hom_comp_expand {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → S) (p : ℕ) : (eval₂Hom f g).comp ↑(expand p) = eval₂Hom f (g ^ p)

@[simp]

theorem MvPolynomial.eval₂_expand {σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → S) (φ : MvPolynomial σ R) (p : ℕ) : eval₂ f g ((expand p) φ) = eval₂ f (g ^ p) φ

@[simp]

theorem MvPolynomial.aeval_comp_expand {σ : Type u_1} {R : Type u_3} [CommSemiring R] {A : Type u_5} [CommSemiring A] [Algebra R A] (f : σ → A) (p : ℕ) : (aeval f).comp (expand p) = aeval (f ^ p)

@[simp]

theorem MvPolynomial.aeval_expand {σ : Type u_1} {R : Type u_3} [CommSemiring R] {A : Type u_5} [CommSemiring A] [Algebra R A] (f : σ → A) (φ : MvPolynomial σ R) (p : ℕ) : (aeval f) ((expand p) φ) = (aeval (f ^ p)) φ

@[simp]

theorem MvPolynomial.eval_expand {σ : Type u_1} {R : Type u_3} [CommSemiring R] (f : σ → R) (φ : MvPolynomial σ R) (p : ℕ) : (eval f) ((expand p) φ) = (eval (f ^ p)) φ

theorem MvPolynomial.expand_mul_eq_comp {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p q : ℕ) : expand (p * q) = (expand p).comp (expand q)

theorem MvPolynomial.expand_mul {σ : Type u_1} {R : Type u_3} [CommSemiring R] (p q : ℕ) (φ : MvPolynomial σ R) : (expand (p * q)) φ = (expand p) ((expand q) φ)

@[simp]

theorem MvPolynomial.coeff_expand_smul {σ : Type u_1} {R : Type u_3} [CommSemiring R] (φ : MvPolynomial σ R) {p : ℕ} (hp : p ≠ 0) (m : σ →₀ ℕ) : coeff (p • m) ((expand p) φ) = coeff m φ

theorem MvPolynomial.support_expand_subset {σ : Type u_1} {R : Type u_3} [CommSemiring R] [DecidableEq σ] (φ : MvPolynomial σ R) (p : ℕ) : ((expand p) φ).support ⊆ Finset.image (fun (x : σ →₀ ℕ) => p • x) φ.support

theorem MvPolynomial.coeff_expand_of_not_dvd {σ : Type u_1} {R : Type u_3} [CommSemiring R] (φ : MvPolynomial σ R) {p : ℕ} {m : σ →₀ ℕ} {i : σ} (h : ¬p ∣ m i) : coeff m ((expand p) φ) = 0

theorem MvPolynomial.support_expand {σ : Type u_1} {R : Type u_3} [CommSemiring R] [DecidableEq σ] (φ : MvPolynomial σ R) {p : ℕ} (hp : p ≠ 0) : ((expand p) φ).support = Finset.image (fun (x : σ →₀ ℕ) => p • x) φ.support