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Mathlib.RingTheory.MvPolynomial.Tower

Algebra towers for multivariate polynomial #

This file proves some basic results about the algebra tower structure for the type MvPolynomial σ R.

This structure itself is provided elsewhere as MvPolynomial.isScalarTower

When you update this file, you can also try to make a corresponding update in RingTheory.Polynomial.Tower.

theorem MvPolynomial.aeval_map_algebraMap {R : Type u_1} (A : Type u_2) {B : Type u_3} {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : σB) (p : MvPolynomial σ R) :
theorem MvPolynomial.aeval_algebraMap_apply {R : Type u_1} {A : Type u_2} (B : Type u_3) {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : σA) (p : MvPolynomial σ R) :
↑(MvPolynomial.aeval (↑(algebraMap A B) x)) p = ↑(algebraMap A B) (↑(MvPolynomial.aeval x) p)
theorem MvPolynomial.aeval_algebraMap_eq_zero_iff {R : Type u_1} {A : Type u_2} (B : Type u_3) {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] [NoZeroSMulDivisors A B] [Nontrivial B] (x : σA) (p : MvPolynomial σ R) :
↑(MvPolynomial.aeval (↑(algebraMap A B) x)) p = 0 ↑(MvPolynomial.aeval x) p = 0
theorem MvPolynomial.aeval_algebraMap_eq_zero_iff_of_injective {R : Type u_1} {A : Type u_2} (B : Type u_3) {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : σA} {p : MvPolynomial σ R} (h : Function.Injective ↑(algebraMap A B)) :
↑(MvPolynomial.aeval (↑(algebraMap A B) x)) p = 0 ↑(MvPolynomial.aeval x) p = 0
@[simp]
theorem Subalgebra.mvPolynomial_aeval_coe {R : Type u_1} {A : Type u_2} {σ : Type u_4} [CommSemiring R] [CommSemiring A] [Algebra R A] (S : Subalgebra R A) (x : σ{ x // x S }) (p : MvPolynomial σ R) :
↑(MvPolynomial.aeval fun i => ↑(x i)) p = ↑(↑(MvPolynomial.aeval x) p)