Documentation

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} [] (p : ) :

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

See also Polynomial.expand.

Equations
Instances For
theorem MvPolynomial.expand_C {σ : Type u_1} {R : Type u_3} [] (p : ) (r : R) :
(MvPolynomial.C r) = MvPolynomial.C r
@[simp]
theorem MvPolynomial.expand_X {σ : Type u_1} {R : Type u_3} [] (p : ) (i : σ) :
=
@[simp]
theorem MvPolynomial.expand_monomial {σ : Type u_1} {R : Type u_3} [] (p : ) (d : σ →₀ ) (r : R) :
( r) = MvPolynomial.C r * id.support, ( ^ p) ^ d i
theorem MvPolynomial.expand_one_apply {σ : Type u_1} {R : Type u_3} [] (f : ) :
f = f
@[simp]
theorem MvPolynomial.expand_one {σ : Type u_1} {R : Type u_3} [] :
theorem MvPolynomial.expand_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [] (p : ) (f : σ) :
.comp = MvPolynomial.bind₁ fun (i : σ) => (f i)
theorem MvPolynomial.expand_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [] (p : ) (f : σ) (φ : ) :
( φ) = (MvPolynomial.bind₁ fun (i : σ) => (f i)) φ
@[simp]
theorem MvPolynomial.map_expand {σ : Type u_1} {R : Type u_3} {S : Type u_4} [] [] (f : R →+* S) (p : ) (φ : ) :
( φ) = ( φ)
@[simp]
theorem MvPolynomial.rename_expand {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [] (f : στ) (p : ) (φ : ) :
( φ) = ( φ)
@[simp]
theorem MvPolynomial.rename_comp_expand {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [] (f : στ) (p : ) :
.comp = .comp