Scaling the roots of a polynomial

This file defines scaleRoots p s for a polynomial p in one variable and a ring element s to be the polynomial with root r * s for each root r of p and proves some basic results about it.

noncomputable def Polynomial.scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) : Polynomial R

scaleRoots p s is a polynomial with root r * s for each root r of p.

Equations

• p.scaleRoots s = ∑ i ∈ p.support, (Polynomial.monomial i) (p.coeff i * s ^ (p.natDegree - i))

@[simp]

theorem Polynomial.coeff_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) (i : ℕ) : (p.scaleRoots s).coeff i = p.coeff i * s ^ (p.natDegree - i)

theorem Polynomial.coeff_scaleRoots_natDegree {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) : (p.scaleRoots s).coeff p.natDegree = p.leadingCoeff

@[simp]

theorem Polynomial.zero_scaleRoots {R : Type u_1} [Semiring R] (s : R) : scaleRoots 0 s = 0

theorem Polynomial.scaleRoots_ne_zero {R : Type u_1} [Semiring R] {p : Polynomial R} (hp : p ≠ 0) (s : R) : p.scaleRoots s ≠ 0

theorem Polynomial.support_scaleRoots_le {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) : (p.scaleRoots s).support ⊆ p.support

theorem Polynomial.support_scaleRoots_eq {R : Type u_1} [Semiring R] (p : Polynomial R) {s : R} (hs : s ∈ nonZeroDivisors R) : (p.scaleRoots s).support = p.support

@[simp]

theorem Polynomial.degree_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) {s : R} : (p.scaleRoots s).degree = p.degree

@[simp]

theorem Polynomial.natDegree_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (s : R) : (p.scaleRoots s).natDegree = p.natDegree

@[simp]

theorem Polynomial.leadingCoeff_scaleRoots {R : Type u_1} [Semiring R] (p : Polynomial R) (r : R) : (p.scaleRoots r).leadingCoeff = p.leadingCoeff

theorem Polynomial.monic_scaleRoots_iff {R : Type u_1} [Semiring R] {p : Polynomial R} (s : R) : (p.scaleRoots s).Monic ↔ p.Monic

theorem Polynomial.map_scaleRoots {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (p : Polynomial R) (x : R) (f : R →+* S) (h : f p.leadingCoeff ≠ 0) : map f (p.scaleRoots x) = (map f p).scaleRoots (f x)

@[simp]

theorem Polynomial.scaleRoots_C {R : Type u_1} [Semiring R] (r c : R) : (C c).scaleRoots r = C c

@[simp]

theorem Polynomial.scaleRoots_one {R : Type u_1} [Semiring R] (p : Polynomial R) : p.scaleRoots 1 = p

@[simp]

theorem Polynomial.scaleRoots_zero {R : Type u_1} [Semiring R] (p : Polynomial R) : p.scaleRoots 0 = p.leadingCoeff • X ^ p.natDegree

@[simp]

theorem Polynomial.one_scaleRoots {R : Type u_1} [Semiring R] (r : R) : scaleRoots 1 r = 1

@[simp]

theorem Polynomial.X_add_C_scaleRoots {R : Type u_1} [Semiring R] (r s : R) : (X + C r).scaleRoots s = X + C (r * s)

theorem Polynomial.scaleRoots_eval₂_mul_of_commute {S : Type u_2} {A : Type u_3} [Semiring S] [Semiring A] {p : Polynomial S} (f : S →+* A) (a : A) (s : S) (hsa : Commute (f s) a) (hf : ∀ (s₁ s₂ : S), Commute (f s₁) (f s₂)) : eval₂ f (f s * a) (p.scaleRoots s) = f s ^ p.natDegree * eval₂ f a p

theorem Polynomial.scaleRoots_eval₂_mul {R : Type u_1} {S : Type u_2} [Semiring S] [CommSemiring R] {p : Polynomial S} (f : S →+* R) (r : R) (s : S) : eval₂ f (f s * r) (p.scaleRoots s) = f s ^ p.natDegree * eval₂ f r p

theorem Polynomial.scaleRoots_eval₂_eq_zero {R : Type u_1} {S : Type u_2} [Semiring S] [CommSemiring R] {p : Polynomial S} (f : S →+* R) {r : R} {s : S} (hr : eval₂ f r p = 0) : eval₂ f (f s * r) (p.scaleRoots s) = 0

theorem Polynomial.scaleRoots_eval_mul {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r s : R) : eval (s * r) (p.scaleRoots s) = s ^ p.natDegree * eval r p

theorem Polynomial.scaleRoots_aeval_smul {R : Type u_1} [CommSemiring R] {S : Type u_5} [CommSemiring S] [Algebra S R] {p : Polynomial S} (r : R) (s : S) : (aeval (s • r)) (p.scaleRoots s) = s ^ p.natDegree • (aeval r) p

theorem Polynomial.scaleRoots_aeval_eq_zero {R : Type u_1} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] {p : Polynomial R} {a : A} {r : R} (ha : (aeval a) p = 0) : (aeval ((algebraMap R A) r * a)) (p.scaleRoots r) = 0

theorem Polynomial.scaleRoots_eval₂_eq_zero_of_eval₂_div_eq_zero {S : Type u_2} {K : Type u_4} [Semiring S] [Field K] {p : Polynomial S} {f : S →+* K} (hf : Function.Injective ⇑f) {r s : S} (hr : eval₂ f (f r / f s) p = 0) (hs : s ∈ nonZeroDivisors S) : eval₂ f (f r) (p.scaleRoots s) = 0

theorem Polynomial.scaleRoots_aeval_eq_zero_of_aeval_div_eq_zero {R : Type u_1} {K : Type u_4} [CommSemiring R] [Field K] [Algebra R K] (inj : Function.Injective ⇑(algebraMap R K)) {p : Polynomial R} {r s : R} (hr : (aeval ((algebraMap R K) r / (algebraMap R K) s)) p = 0) (hs : s ∈ nonZeroDivisors R) : (aeval ((algebraMap R K) r)) (p.scaleRoots s) = 0

@[simp]

theorem Polynomial.scaleRoots_mul {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r s : R) : p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s

theorem Polynomial.mul_scaleRoots {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) : r ^ (p.natDegree + q.natDegree - (p * q).natDegree) • (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r

Multiplication and scaleRoots commute up to a power of r. The factor disappears if we assume that the product of the leading coeffs does not vanish. See Polynomial.mul_scaleRoots'.

theorem Polynomial.mul_scaleRoots' {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (h : p.leadingCoeff * q.leadingCoeff ≠ 0) : (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r

theorem Polynomial.mul_scaleRoots_of_noZeroDivisors {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) [NoZeroDivisors R] : (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r

theorem Polynomial.pow_scaleRoots' {R : Type u_1} [CommSemiring R] (p : Polynomial R) (r : R) (n : ℕ) (hp : p.leadingCoeff ^ n ≠ 0) : (p ^ n).scaleRoots r = p.scaleRoots r ^ n

theorem Polynomial.pow_scaleRoots_of_isReduced {R : Type u_1} [CommSemiring R] [IsReduced R] (p : Polynomial R) (r : R) (n : ℕ) : (p ^ n).scaleRoots r = p.scaleRoots r ^ n

theorem Polynomial.add_scaleRoots_of_natDegree_eq {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (h : p.natDegree = q.natDegree) : r ^ (p.natDegree - (p + q).natDegree) • (p + q).scaleRoots r = p.scaleRoots r + q.scaleRoots r

theorem Polynomial.scaleRoots_dvd' {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r) (hpq : p ∣ q) : p.scaleRoots r ∣ q.scaleRoots r

theorem Polynomial.scaleRoots_dvd {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} [NoZeroDivisors R] (hpq : p ∣ q) : p.scaleRoots r ∣ q.scaleRoots r

theorem Dvd.dvd.scaleRoots {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} [NoZeroDivisors R] (hpq : p ∣ q) : p.scaleRoots r ∣ q.scaleRoots r

Alias of Polynomial.scaleRoots_dvd.

theorem Polynomial.scaleRoots_dvd_iff {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r) : p.scaleRoots r ∣ q.scaleRoots r ↔ p ∣ q

theorem IsUnit.scaleRoots_dvd_iff {R : Type u_1} [CommSemiring R] (p q : Polynomial R) {r : R} (hr : IsUnit r) : p.scaleRoots r ∣ q.scaleRoots r ↔ p ∣ q

Alias of Polynomial.scaleRoots_dvd_iff.

theorem Polynomial.isCoprime_scaleRoots {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (hr : IsUnit r) (h : IsCoprime p q) : IsCoprime (p.scaleRoots r) (q.scaleRoots r)

theorem IsCoprime.scaleRoots {R : Type u_1} [CommSemiring R] (p q : Polynomial R) (r : R) (hr : IsUnit r) (h : IsCoprime p q) : IsCoprime (p.scaleRoots r) (q.scaleRoots r)

Alias of Polynomial.isCoprime_scaleRoots.

theorem Polynomial.Splits.scaleRoots {R : Type u_1} [CommSemiring R] {p : Polynomial R} (hp : p.Splits) (r : R) : (p.scaleRoots r).Splits

@[simp]

theorem Polynomial.X_sub_C_scaleRoots {R : Type u_1} [Ring R] (r s : R) : (X - C r).scaleRoots s = X - C (r * s)

theorem Polynomial.rootMultiplicity_scaleRoots {R : Type u_1} [CommRing R] (p : Polynomial R) {r a : R} (hr : IsLeftRegular r) : rootMultiplicity (r * a) (p.scaleRoots r) = rootMultiplicity a p

theorem Polynomial.roots_scaleRoots {R : Type u_1} [CommRing R] [IsDomain R] (p : Polynomial R) {r : R} (hr : IsUnit r) : (p.scaleRoots r).roots = Multiset.map (fun (x : R) => r * x) p.roots