mathlib documentation

data.mv_polynomial.expand

Expand multivariate polynomials #

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

Main declaration #

def mv_polynomial.expand {σ : Type u_1} {R : Type u_3} [comm_semiring R] (p : ) :

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

See also polynomial.expand.

Equations
@[simp]
theorem mv_polynomial.expand_C {σ : Type u_1} {R : Type u_3} [comm_semiring R] (p : ) (r : R) :
@[simp]
theorem mv_polynomial.expand_X {σ : Type u_1} {R : Type u_3} [comm_semiring R] (p : ) (i : σ) :
@[simp]
theorem mv_polynomial.expand_monomial {σ : Type u_1} {R : Type u_3} [comm_semiring R] (p : ) (d : σ →₀ ) (r : R) :
theorem mv_polynomial.expand_one_apply {σ : Type u_1} {R : Type u_3} [comm_semiring R] (f : mv_polynomial σ R) :
@[simp]
theorem mv_polynomial.expand_one {σ : Type u_1} {R : Type u_3} [comm_semiring R] :
theorem mv_polynomial.expand_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (p : ) (f : σ → mv_polynomial τ R) :
theorem mv_polynomial.expand_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (p : ) (f : σ → mv_polynomial τ R) (φ : mv_polynomial σ R) :
@[simp]
theorem mv_polynomial.map_expand {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* S) (p : ) (φ : mv_polynomial σ R) :
@[simp]
theorem mv_polynomial.rename_expand {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → τ) (p : ) (φ : mv_polynomial σ R) :
@[simp]
theorem mv_polynomial.rename_comp_expand {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → τ) (p : ) :