mathlib documentation

data.polynomial.algebra_map

Theory of univariate polynomials #

We show that A[X] is an R-algebra when A is an R-algebra. We promote eval₂ to an algebra hom in aeval.

@[protected, instance]
noncomputable def polynomial.algebra_of_algebra {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] :

Note that this instance also provides algebra R R[X].

Equations

When we have [comm_semiring R], the function C is the same as algebra_map R R[X].

(But note that C is defined when R is not necessarily commutative, in which case algebra_map is not available.)

@[ext]
theorem polynomial.alg_hom_ext' {R : Type u} {A' : Type u_1} {B' : Type u_2} [comm_semiring A'] [semiring B'] [comm_semiring R] [algebra R A'] [algebra R B'] {f g : polynomial A' →ₐ[R] B'} (h₁ : f.comp (is_scalar_tower.to_alg_hom R A' (polynomial A')) = g.comp (is_scalar_tower.to_alg_hom R A' (polynomial A'))) (h₂ : f polynomial.X = g polynomial.X) :
f = g

Extensionality lemma for algebra maps out of A'[X] over a smaller base ring than A'

Algebra isomorphism between R[X] and add_monoid_algebra R ℕ. This is just an implementation detail, but it can be useful to transfer results from finsupp to polynomials.

Equations
@[protected, instance]
@[simp]
theorem polynomial.alg_hom_eval₂_algebra_map {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (p : polynomial R) (f : A →ₐ[R] B) (a : A) :
@[simp]
@[simp]
noncomputable def polynomial.aeval {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) :

Given a valuation x of the variable in an R-algebra A, aeval R A x is the unique R-algebra homomorphism from R[X] to A sending X to x.

This is a stronger variant of the linear map polynomial.leval.

Equations
@[ext]
theorem polynomial.alg_hom_ext {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] {f g : polynomial R →ₐ[R] A} (h : f polynomial.X = g polynomial.X) :
f = g
theorem polynomial.aeval_def {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) (p : polynomial R) :
@[simp]
theorem polynomial.aeval_zero {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) :
@[simp]
theorem polynomial.aeval_X {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) :
@[simp]
theorem polynomial.aeval_C {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) (r : R) :
@[simp]
theorem polynomial.aeval_monomial {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) {n : } {r : R} :
@[simp]
theorem polynomial.aeval_X_pow {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) {n : } :
@[simp]
theorem polynomial.aeval_add {R : Type u} {A : Type z} [comm_semiring R] {p q : polynomial R} [semiring A] [algebra R A] (x : A) :
@[simp]
theorem polynomial.aeval_one {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) :
@[simp]
theorem polynomial.aeval_bit0 {R : Type u} {A : Type z} [comm_semiring R] {p : polynomial R} [semiring A] [algebra R A] (x : A) :
@[simp]
theorem polynomial.aeval_bit1 {R : Type u} {A : Type z} [comm_semiring R] {p : polynomial R} [semiring A] [algebra R A] (x : A) :
@[simp]
theorem polynomial.aeval_nat_cast {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (x : A) (n : ) :
theorem polynomial.aeval_mul {R : Type u} {A : Type z} [comm_semiring R] {p q : polynomial R} [semiring A] [algebra R A] (x : A) :
theorem polynomial.aeval_comp {R : Type u} [comm_semiring R] {p q : polynomial R} {A : Type u_1} [comm_semiring A] [algebra R A] (x : A) :
theorem polynomial.aeval_alg_hom {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] {B : Type u_3} [semiring B] [algebra R B] (f : A →ₐ[R] B) (x : A) :
theorem polynomial.eval_unique {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] (φ : polynomial R →ₐ[R] A) (p : polynomial R) :
theorem polynomial.aeval_alg_hom_apply {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] {B : Type u_3} [semiring B] [algebra R B] {F : Type u_1} [alg_hom_class F R A B] (f : F) (x : A) (p : polynomial R) :
theorem polynomial.aeval_alg_equiv {R : Type u} {A : Type z} [comm_semiring R] [semiring A] [algebra R A] {B : Type u_3} [semiring B] [algebra R B] (f : A ≃ₐ[R] B) (x : A) :
@[simp]
theorem polynomial.aeval_fn_apply {R : Type u} [comm_semiring R] {X : Type u_1} (g : polynomial R) (f : X R) (x : X) :
@[norm_cast]
theorem polynomial.aeval_subalgebra_coe {R : Type u} [comm_semiring R] (g : polynomial R) {A : Type u_1} [semiring A] [algebra R A] (s : subalgebra R A) (f : s) :
theorem polynomial.map_aeval_eq_aeval_map {R : Type u} [comm_semiring R] {S : Type u_1} {T : Type u_2} {U : Type u_3} [comm_semiring S] [comm_semiring T] [semiring U] [algebra R S] [algebra T U] {φ : R →+* T} {ψ : S →+* U} (h : (algebra_map T U).comp φ = ψ.comp (algebra_map R S)) (p : polynomial R) (a : S) :
theorem polynomial.aeval_eq_zero_of_dvd_aeval_eq_zero {S : Type v} {T : Type w} [comm_semiring S] [comm_semiring T] [algebra S T] {p q : polynomial S} (h₁ : p q) {a : T} (h₂ : (polynomial.aeval a) p = 0) :
theorem polynomial.aeval_eq_sum_range {R : Type u} {S : Type v} [comm_semiring R] [semiring S] [algebra R S] {p : polynomial R} (x : S) :
(polynomial.aeval x) p = (finset.range (p.nat_degree + 1)).sum (λ (i : ), p.coeff i x ^ i)
theorem polynomial.aeval_eq_sum_range' {R : Type u} {S : Type v} [comm_semiring R] [semiring S] [algebra R S] {p : polynomial R} {n : } (hn : p.nat_degree < n) (x : S) :
(polynomial.aeval x) p = (finset.range n).sum (λ (i : ), p.coeff i x ^ i)
theorem polynomial.is_root_of_eval₂_map_eq_zero {R : Type u} {S : Type v} [comm_semiring R] {p : polynomial R} [semiring S] {f : R →+* S} (hf : function.injective f) {r : R} :
noncomputable def polynomial.aeval_tower {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (f : R →ₐ[S] A') (x : A') :

Version of aeval for defining algebra homs out of R[X] over a smaller base ring than R.

Equations
@[simp]
theorem polynomial.aeval_tower_X {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') :
@[simp]
theorem polynomial.aeval_tower_C {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') (x : R) :
@[simp]
theorem polynomial.aeval_tower_comp_C {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') :
@[simp]
theorem polynomial.aeval_tower_algebra_map {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') (x : R) :
@[simp]
theorem polynomial.aeval_tower_comp_algebra_map {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') :
theorem polynomial.aeval_tower_to_alg_hom {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') (x : R) :
@[simp]
theorem polynomial.aeval_tower_comp_to_alg_hom {R : Type u} {S : Type v} {A' : Type u_1} [comm_semiring A'] [comm_semiring R] [comm_semiring S] [algebra S R] [algebra S A'] (g : R →ₐ[S] A') (y : A') :
theorem polynomial.dvd_term_of_dvd_eval_of_dvd_terms {S : Type v} [comm_ring S] {z p : S} {f : polynomial S} (i : ) (dvd_eval : p polynomial.eval z f) (dvd_terms : (j : ), j i p f.coeff j * z ^ j) :
p f.coeff i * z ^ i
theorem polynomial.dvd_term_of_is_root_of_dvd_terms {S : Type v} [comm_ring S] {r p : S} {f : polynomial S} (i : ) (hr : f.is_root r) (h : (j : ), j i p f.coeff j * r ^ j) :
p f.coeff i * r ^ i
theorem polynomial.eval_mul_X_sub_C {R : Type u} [ring R] {p : polynomial R} (r : R) :

The evaluation map is not generally multiplicative when the coefficient ring is noncommutative, but nevertheless any polynomial of the form p * (X - monomial 0 r) is sent to zero when evaluated at r.

This is the key step in our proof of the Cayley-Hamilton theorem.

theorem polynomial.aeval_endomorphism {R : Type u} {M : Type u_1} [comm_ring R] [add_comm_group M] [module R M] (f : M →ₗ[R] M) (v : M) (p : polynomial R) :
((polynomial.aeval f) p) v = p.sum (λ (n : ) (b : R), b (f ^ n) v)