# mathlib3documentation

data.mv_polynomial.rename

# Renaming variables of polynomials #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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

## Main declarations #

• mv_polynomial.rename
• mv_polynomial.rename_equiv

## Notation #

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

• σ τ α : Type* (indexing the variables)

• R S : Type* [comm_semiring R] [comm_semiring S] (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 mv_polynomial σ R which mathematicians might call X^s

• r : R elements of the coefficient ring

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

• p : mv_polynomial σ α

noncomputable def mv_polynomial.rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (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} (f : σ τ) (r : R) :
@[simp]
theorem mv_polynomial.rename_X {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (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} (f : R →+* S) (g : σ τ) (p : R) :
p) = ( p)
@[simp]
theorem mv_polynomial.rename_rename {σ : Type u_1} {τ : Type u_2} {α : Type u_3} {R : Type u_4} (f : σ τ) (g : τ α) (p : R) :
@[simp]
theorem mv_polynomial.rename_id {σ : Type u_1} {R : Type u_4} (p : R) :
= p
theorem mv_polynomial.rename_monomial {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (f : σ τ) (d : σ →₀ ) (r : R) :
r) =
theorem mv_polynomial.rename_eq {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (f : σ τ) (p : R) :
theorem mv_polynomial.rename_injective {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (f : σ τ) (hf : function.injective f) :
noncomputable def mv_polynomial.kill_compl {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {f : σ τ} (hf : function.injective f) :

Given a function between sets of variables f : σ → τ that is injective with proof hf, kill_compl hf is the alg_hom from R[τ] to R[σ] that is left inverse to rename f : R[σ] → R[τ] and sends the variables in the complement of the range of f to 0.

Equations
theorem mv_polynomial.kill_compl_comp_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {f : σ τ} (hf : function.injective f) :
= R)
@[simp]
theorem mv_polynomial.kill_compl_rename_app {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {f : σ τ} (hf : function.injective f) (p : R) :
p) = p
@[simp]
theorem mv_polynomial.rename_equiv_apply {σ : Type u_1} {τ : Type u_2} (R : Type u_4) (f : σ τ) (ᾰ : R) :
=
noncomputable def mv_polynomial.rename_equiv {σ : Type u_1} {τ : Type u_2} (R : Type u_4) (f : σ τ) :

mv_polynomial.rename e is an equivalence when e is.

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

Every polynomial is a polynomial in finitely many variables.

theorem mv_polynomial.exists_finset_rename₂ {σ : Type u_1} {R : Type u_4} (p₁ p₂ : R) :
(s : finset σ) (q₁ q₂ : R), p₁ = p₂ =

exists_finset_rename for two polyonomials at once: for any two polynomials p₁, p₂ in a polynomial semiring R[σ] of possibly infinitely many variables, exists_finset_rename₂ yields a finite subset s of σ such that both p₁ and p₂ are contained in the polynomial semiring R[s] of finitely many variables.

theorem mv_polynomial.exists_fin_rename {σ : Type u_1} {R : Type u_4} (p : R) :
(n : ) (f : fin n σ) (hf : (q : mv_polynomial (fin n) R), p =

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} (f : σ τ) (c : →+* R) (g : τ R) (p : ) :
(g f) p = p)
@[simp]
theorem mv_polynomial.coeff_rename_map_domain {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (f : σ τ) (hf : function.injective f) (φ : R) (d : σ →₀ ) :
φ) =
theorem mv_polynomial.coeff_rename_eq_zero {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (f : σ τ) (φ : R) (d : τ →₀ ) (h : (u : σ →₀ ), = d = 0) :
φ) = 0
theorem mv_polynomial.coeff_rename_ne_zero {σ : Type u_1} {τ : Type u_2} {R : Type u_4} (f : σ τ) (φ : R) (d : τ →₀ ) (h : φ) 0) :
(u : σ →₀ ), = d 0
@[simp]
theorem mv_polynomial.constant_coeff_rename {σ : Type u_1} {R : Type u_4} {τ : Type u_2} (f : σ τ) (φ : R) :
theorem mv_polynomial.support_rename_of_injective {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {p : R} {f : σ τ} [decidable_eq τ] (h : function.injective f) :
p).support =