Documentation

Mathlib.Algebra.MvPolynomial.Variables

Variables of polynomials #

This file establishes many results about the variable sets of a multivariate polynomial.

The variable set of a polynomial $P \in R[X]$ is a Finset containing each $x \in X$ that appears in a monomial in $P$.

Main declarations #

Notation #

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

vars #

def MvPolynomial.vars {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) :

vars p is the set of variables appearing in the polynomial p

Equations
  • p.vars = p.degrees.toFinset
Instances For
    theorem MvPolynomial.vars_def {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (p : MvPolynomial σ R) :
    p.vars = p.degrees.toFinset
    @[simp]
    theorem MvPolynomial.vars_0 {R : Type u} {σ : Type u_1} [CommSemiring R] :
    @[simp]
    theorem MvPolynomial.vars_monomial {R : Type u} {σ : Type u_1} {r : R} {s : σ →₀ } [CommSemiring R] (h : r 0) :
    ((MvPolynomial.monomial s) r).vars = s.support
    @[simp]
    theorem MvPolynomial.vars_C {R : Type u} {σ : Type u_1} {r : R} [CommSemiring R] :
    (MvPolynomial.C r).vars =
    @[simp]
    theorem MvPolynomial.vars_X {R : Type u} {σ : Type u_1} {n : σ} [CommSemiring R] [Nontrivial R] :
    (MvPolynomial.X n).vars = {n}
    theorem MvPolynomial.mem_vars {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} (i : σ) :
    i p.vars dp.support, i d.support
    theorem MvPolynomial.mem_support_not_mem_vars_zero {R : Type u} {σ : Type u_1} [CommSemiring R] {f : MvPolynomial σ R} {x : σ →₀ } (H : x f.support) {v : σ} (h : vf.vars) :
    x v = 0
    theorem MvPolynomial.vars_add_subset {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (p : MvPolynomial σ R) (q : MvPolynomial σ R) :
    (p + q).vars p.vars q.vars
    theorem MvPolynomial.vars_add_of_disjoint {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {q : MvPolynomial σ R} [DecidableEq σ] (h : Disjoint p.vars q.vars) :
    (p + q).vars = p.vars q.vars
    theorem MvPolynomial.vars_mul {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (φ : MvPolynomial σ R) (ψ : MvPolynomial σ R) :
    (φ * ψ).vars φ.vars ψ.vars
    @[simp]
    theorem MvPolynomial.vars_pow {R : Type u} {σ : Type u_1} [CommSemiring R] (φ : MvPolynomial σ R) (n : ) :
    (φ ^ n).vars φ.vars
    theorem MvPolynomial.vars_prod {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} [DecidableEq σ] {s : Finset ι} (f : ιMvPolynomial σ R) :
    (∏ is, f i).vars s.biUnion fun (i : ι) => (f i).vars

    The variables of the product of a family of polynomials are a subset of the union of the sets of variables of each polynomial.

    theorem MvPolynomial.vars_C_mul {σ : Type u_1} {A : Type u_3} [CommRing A] [NoZeroDivisors A] (a : A) (ha : a 0) (φ : MvPolynomial σ A) :
    (MvPolynomial.C a * φ).vars = φ.vars
    theorem MvPolynomial.vars_sum_subset {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (t : Finset ι) (φ : ιMvPolynomial σ R) [DecidableEq σ] :
    (∑ it, φ i).vars t.biUnion fun (i : ι) => (φ i).vars
    theorem MvPolynomial.vars_sum_of_disjoint {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (t : Finset ι) (φ : ιMvPolynomial σ R) [DecidableEq σ] (h : Pairwise (Disjoint on fun (i : ι) => (φ i).vars)) :
    (∑ it, φ i).vars = t.biUnion fun (i : ι) => (φ i).vars
    theorem MvPolynomial.vars_map {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) [CommSemiring S] (f : R →+* S) :
    ((MvPolynomial.map f) p).vars p.vars
    theorem MvPolynomial.vars_map_of_injective {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) [CommSemiring S] {f : R →+* S} (hf : Function.Injective f) :
    ((MvPolynomial.map f) p).vars = p.vars
    theorem MvPolynomial.vars_monomial_single {R : Type u} {σ : Type u_1} [CommSemiring R] (i : σ) {e : } {r : R} (he : e 0) (hr : r 0) :
    theorem MvPolynomial.vars_eq_support_biUnion_support {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) [DecidableEq σ] :
    p.vars = p.support.biUnion Finsupp.support

    vars and eval #

    theorem MvPolynomial.eval₂Hom_eq_constantCoeff_of_vars {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] (f : R →+* S) {g : σS} {p : MvPolynomial σ R} (hp : ip.vars, g i = 0) :
    (MvPolynomial.eval₂Hom f g) p = f (MvPolynomial.constantCoeff p)
    theorem MvPolynomial.aeval_eq_constantCoeff_of_vars {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] [Algebra R S] {g : σS} {p : MvPolynomial σ R} (hp : ip.vars, g i = 0) :
    (MvPolynomial.aeval g) p = (algebraMap R S) (MvPolynomial.constantCoeff p)
    theorem MvPolynomial.eval₂Hom_congr' {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] {f₁ : R →+* S} {f₂ : R →+* S} {g₁ : σS} {g₂ : σS} {p₁ : MvPolynomial σ R} {p₂ : MvPolynomial σ R} :
    f₁ = f₂(∀ ip₁.vars, i p₂.varsg₁ i = g₂ i)p₁ = p₂(MvPolynomial.eval₂Hom f₁ g₁) p₁ = (MvPolynomial.eval₂Hom f₂ g₂) p₂
    theorem MvPolynomial.hom_congr_vars {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] {f₁ : MvPolynomial σ R →+* S} {f₂ : MvPolynomial σ R →+* S} {p₁ : MvPolynomial σ R} {p₂ : MvPolynomial σ R} (hC : f₁.comp MvPolynomial.C = f₂.comp MvPolynomial.C) (hv : ip₁.vars, i p₂.varsf₁ (MvPolynomial.X i) = f₂ (MvPolynomial.X i)) (hp : p₁ = p₂) :
    f₁ p₁ = f₂ p₂

    If f₁ and f₂ are ring homs out of the polynomial ring and p₁ and p₂ are polynomials, then f₁ p₁ = f₂ p₂ if p₁ = p₂ and f₁ and f₂ are equal on R and on the variables of p₁.

    theorem MvPolynomial.exists_rename_eq_of_vars_subset_range {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] (p : MvPolynomial σ R) (f : τσ) (hfi : Function.Injective f) (hf : p.vars Set.range f) :
    ∃ (q : MvPolynomial τ R), (MvPolynomial.rename f) q = p
    theorem MvPolynomial.vars_rename {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] [DecidableEq τ] (f : στ) (φ : MvPolynomial σ R) :
    ((MvPolynomial.rename f) φ).vars Finset.image f φ.vars
    theorem MvPolynomial.mem_vars_rename {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] (f : στ) (φ : MvPolynomial σ R) {j : τ} (h : j ((MvPolynomial.rename f) φ).vars) :
    iφ.vars, f i = j
    theorem MvPolynomial.supDegree_toLex_C {R : Type u} {σ : Type u_1} [CommSemiring R] [LinearOrder σ] (r : R) :
    AddMonoidAlgebra.supDegree (⇑toLex) (MvPolynomial.C r) = 0
    theorem MvPolynomial.leadingCoeff_toLex_C {R : Type u} {σ : Type u_1} [CommSemiring R] [LinearOrder σ] (r : R) :
    AddMonoidAlgebra.leadingCoeff (⇑toLex) (MvPolynomial.C r) = r