Theory of univariate polynomials

We prove basic results about univariate polynomials.

theorem Polynomial.natDegree_pos_of_aeval_root {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : Polynomial R} (hp : p ≠ 0) {z : S} (hz : (Polynomial.aeval z) p = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0) : 0 < p.natDegree

theorem Polynomial.degree_pos_of_aeval_root {R : Type u} {S : Type v} [CommRing R] [Semiring S] [Algebra R S] {p : Polynomial R} (hp : p ≠ 0) {z : S} (hz : (Polynomial.aeval z) p = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0) : 0 < p.degree

theorem Polynomial.smul_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (c : R) (p : Polynomial R) : c • p %ₘ q = c • (p %ₘ q)

def Polynomial.modByMonicHom {R : Type u} [CommRing R] (q : Polynomial R) : Polynomial R →ₗ[R] Polynomial R

_ %ₘ q as an R-linear map.

Equations

• q.modByMonicHom = { toFun := fun (p : Polynomial R) => p %ₘ q, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem Polynomial.modByMonicHom_apply {R : Type u} [CommRing R] (q p : Polynomial R) : q.modByMonicHom p = p %ₘ q

theorem Polynomial.mem_ker_modByMonic {R : Type u} [CommRing R] {q : Polynomial R} (hq : q.Monic) {p : Polynomial R} : p ∈ LinearMap.ker q.modByMonicHom ↔ q ∣ p

@[simp]

theorem Polynomial.ker_modByMonicHom {R : Type u} [CommRing R] {q : Polynomial R} (hq : q.Monic) : LinearMap.ker q.modByMonicHom = Submodule.restrictScalars R (Ideal.span {q})

theorem Polynomial.aeval_modByMonic_eq_self_of_root {R : Type u} {S : Type v} [CommRing R] [Ring S] [Algebra R S] {p q : Polynomial R} (hq : q.Monic) {x : S} (hx : (Polynomial.aeval x) q = 0) : (Polynomial.aeval x) (p %ₘ q) = (Polynomial.aeval x) p

theorem Polynomial.trailingDegree_mul {R : Type u} [Semiring R] [NoZeroDivisors R] {p q : Polynomial R} : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree

theorem Polynomial.rootMultiplicity_eq_rootMultiplicity {R : Type u} [CommRing R] {p : Polynomial R} {t : R} : Polynomial.rootMultiplicity t p = Polynomial.rootMultiplicity 0 (p.comp (Polynomial.X + Polynomial.C t))

theorem Polynomial.rootMultiplicity_eq_natTrailingDegree {R : Type u} [CommRing R] {p : Polynomial R} {t : R} : Polynomial.rootMultiplicity t p = (p.comp (Polynomial.X + Polynomial.C t)).natTrailingDegree

See Polynomial.rootMultiplicity_eq_natTrailingDegree' for the special case of t = 0.

theorem Polynomial.Monic.mem_nonZeroDivisors {R : Type u} [CommRing R] {p : Polynomial R} (h : p.Monic) : p ∈ nonZeroDivisors (Polynomial R)

theorem Polynomial.mem_nonZeroDivisors_of_leadingCoeff {R : Type u} [CommRing R] {p : Polynomial R} (h : p.leadingCoeff ∈ nonZeroDivisors R) : p ∈ nonZeroDivisors (Polynomial R)

theorem Polynomial.rootMultiplicity_mul_X_sub_C_pow {R : Type u} [CommRing R] {p : Polynomial R} {a : R} {n : ℕ} (h : p ≠ 0) : Polynomial.rootMultiplicity a (p * (Polynomial.X - Polynomial.C a) ^ n) = Polynomial.rootMultiplicity a p + n

theorem Polynomial.rootMultiplicity_X_sub_C_pow {R : Type u} [CommRing R] [Nontrivial R] (a : R) (n : ℕ) : Polynomial.rootMultiplicity a ((Polynomial.X - Polynomial.C a) ^ n) = n

The multiplicity of a as root of (X - a) ^ n is n.

theorem Polynomial.rootMultiplicity_X_sub_C_self {R : Type u} [CommRing R] [Nontrivial R] {x : R} : Polynomial.rootMultiplicity x (Polynomial.X - Polynomial.C x) = 1

theorem Polynomial.rootMultiplicity_X_sub_C {R : Type u} [CommRing R] [Nontrivial R] [DecidableEq R] {x y : R} : Polynomial.rootMultiplicity x (Polynomial.X - Polynomial.C y) = if x = y then 1 else 0

theorem Polynomial.rootMultiplicity_mul' {R : Type u} [CommRing R] {p q : Polynomial R} {x : R} (hpq : Polynomial.eval x (p /ₘ (Polynomial.X - Polynomial.C x) ^ Polynomial.rootMultiplicity x p) * Polynomial.eval x (q /ₘ (Polynomial.X - Polynomial.C x) ^ Polynomial.rootMultiplicity x q) ≠ 0) : Polynomial.rootMultiplicity x (p * q) = Polynomial.rootMultiplicity x p + Polynomial.rootMultiplicity x q

theorem Polynomial.Monic.neg_one_pow_natDegree_mul_comp_neg_X {R : Type u} [CommRing R] {p : Polynomial R} (hp : p.Monic) : ((-1) ^ p.natDegree * p.comp (-Polynomial.X)).Monic

theorem Polynomial.degree_eq_degree_of_associated {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} (h : Associated p q) : p.degree = q.degree

theorem Polynomial.prime_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) : Prime (Polynomial.X - Polynomial.C r)

theorem Polynomial.prime_X {R : Type u} [CommRing R] [IsDomain R] : Prime Polynomial.X

theorem Polynomial.Monic.prime_of_degree_eq_one {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp1 : p.degree = 1) (hm : p.Monic) : Prime p

theorem Polynomial.irreducible_X_sub_C {R : Type u} [CommRing R] [IsDomain R] (r : R) : Irreducible (Polynomial.X - Polynomial.C r)

theorem Polynomial.irreducible_X {R : Type u} [CommRing R] [IsDomain R] : Irreducible Polynomial.X

theorem Polynomial.Monic.irreducible_of_degree_eq_one {R : Type u} [CommRing R] [IsDomain R] {p : Polynomial R} (hp1 : p.degree = 1) (hm : p.Monic) : Irreducible p

theorem Polynomial.aeval_ne_zero_of_isCoprime {S : Type v} {R : Type u_1} [CommSemiring R] [Nontrivial S] [Semiring S] [Algebra R S] {p q : Polynomial R} (h : IsCoprime p q) (s : S) : (Polynomial.aeval s) p ≠ 0 ∨ (Polynomial.aeval s) q ≠ 0

theorem Polynomial.isCoprime_X_sub_C_of_isUnit_sub {R : Type u_1} [CommRing R] {a b : R} (h : IsUnit (a - b)) : IsCoprime (Polynomial.X - Polynomial.C a) (Polynomial.X - Polynomial.C b)

theorem Polynomial.pairwise_coprime_X_sub_C {K : Type u_1} [Field K] {I : Type v} {s : I → K} (H : Function.Injective s) : Pairwise (IsCoprime on fun (i : I) => Polynomial.X - Polynomial.C (s i))

theorem Polynomial.rootMultiplicity_mul {R : Type u} [CommRing R] [IsDomain R] {p q : Polynomial R} {x : R} (hpq : p * q ≠ 0) : Polynomial.rootMultiplicity x (p * q) = Polynomial.rootMultiplicity x p + Polynomial.rootMultiplicity x q

@[irreducible]

theorem Polynomial.exists_multiset_roots {R : Type u} [CommRing R] [IsDomain R] [DecidableEq R] {p : Polynomial R} : p ≠ 0 → ∃ (s : Multiset R), ↑(Multiset.card s) ≤ p.degree ∧ ∀ (a : R), Multiset.count a s = Polynomial.rootMultiplicity a p