mathlib documentation

data.mv_polynomial.rename

Renaming variables of polynomials #

This file establishes the rename operation on multivariate polynomials, which modifies the set of variables.

Main declarations #

Notation #

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

def mv_polynomial.rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) :

Rename all the variables in a multivariable polynomial.

Equations
@[simp]
theorem mv_polynomial.rename_C {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (r : R) :
@[simp]
theorem mv_polynomial.rename_X {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (i : σ) :
theorem mv_polynomial.map_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : σ → τ) (p : mv_polynomial σ R) :
@[simp]
theorem mv_polynomial.rename_rename {σ : Type u_1} {τ : Type u_2} {α : Type u_3} {R : Type u_4} [comm_semiring R] (f : σ → τ) (g : τ → α) (p : mv_polynomial σ R) :
@[simp]
theorem mv_polynomial.rename_id {σ : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) :
theorem mv_polynomial.rename_monomial {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (d : σ →₀ ) (r : R) :
theorem mv_polynomial.rename_eq {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (p : mv_polynomial σ R) :
theorem mv_polynomial.rename_injective {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (hf : function.injective f) :
@[simp]
theorem mv_polynomial.rename_equiv_apply {σ : Type u_1} {τ : Type u_2} (R : Type u_4) [comm_semiring R] (f : σ τ) (ᾰ : mv_polynomial σ R) :
def mv_polynomial.rename_equiv {σ : Type u_1} {τ : Type u_2} (R : Type u_4) [comm_semiring R] (f : σ τ) :

mv_polynomial.rename e is an equivalence when e is.

Equations
@[simp]
@[simp]
theorem mv_polynomial.rename_equiv_symm {σ : Type u_1} {τ : Type u_2} (R : Type u_4) [comm_semiring R] (f : σ τ) :
@[simp]
theorem mv_polynomial.rename_equiv_trans {σ : Type u_1} {τ : Type u_2} {α : Type u_3} (R : Type u_4) [comm_semiring R] (e : σ τ) (f : τ α) :
theorem mv_polynomial.eval₂_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (k : σ → τ) (g : τ → S) (p : mv_polynomial σ R) :
theorem mv_polynomial.eval₂_hom_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (k : σ → τ) (g : τ → S) (p : mv_polynomial σ R) :
theorem mv_polynomial.aeval_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (k : σ → τ) (g : τ → S) (p : mv_polynomial σ R) [algebra R S] :
theorem mv_polynomial.rename_eval₂ {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (k : σ → τ) (p : mv_polynomial σ R) (g : τ → mv_polynomial σ R) :
theorem mv_polynomial.rename_prodmk_eval₂ {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) (j : τ) (g : σ → mv_polynomial σ R) :
theorem mv_polynomial.eval₂_rename_prodmk {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : σ × τ → S) (i : σ) (p : mv_polynomial τ R) :
theorem mv_polynomial.eval_rename_prodmk {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (g : σ × τ → R) (i : σ) (p : mv_polynomial τ R) :
theorem mv_polynomial.exists_finset_rename {σ : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) :
∃ (s : finset σ) (q : mv_polynomial {x // x s} R), p = (mv_polynomial.rename coe) q

Every polynomial is a polynomial in finitely many variables.

theorem mv_polynomial.exists_fin_rename {σ : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) :
∃ (n : ) (f : fin n → σ) (hf : function.injective f) (q : mv_polynomial (fin n) R), p = (mv_polynomial.rename f) q

Every polynomial is a polynomial in finitely many variables.

theorem mv_polynomial.eval₂_cast_comp {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (c : →+* R) (g : τ → R) (p : mv_polynomial σ ) :
@[simp]
theorem mv_polynomial.coeff_rename_map_domain {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (hf : function.injective f) (φ : mv_polynomial σ R) (d : σ →₀ ) :
theorem mv_polynomial.coeff_rename_eq_zero {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (φ : mv_polynomial σ R) (d : τ →₀ ) (h : ∀ (u : σ →₀ ), finsupp.map_domain f u = dmv_polynomial.coeff u φ = 0) :
theorem mv_polynomial.coeff_rename_ne_zero {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (φ : mv_polynomial σ R) (d : τ →₀ ) (h : mv_polynomial.coeff d ((mv_polynomial.rename f) φ) 0) :
@[simp]
theorem mv_polynomial.constant_coeff_rename {σ : Type u_1} {R : Type u_4} [comm_semiring R] {τ : Type u_2} (f : σ → τ) (φ : mv_polynomial σ R) :