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

• Polynomial.scaleRoots p s = Finset.sum (Polynomial.support p) fun (i : ℕ) => (Polynomial.monomial i) (Polynomial.coeff p i * s ^ (Polynomial.natDegree p - i))

@[simp]

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

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

@[simp]

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

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

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

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

@[simp]

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

@[simp]

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

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

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 (Polynomial.leadingCoeff p) ≠ 0) : Polynomial.map f (Polynomial.scaleRoots p x) = Polynomial.scaleRoots (Polynomial.map f p) (f x)

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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)) : Polynomial.eval₂ f (f s * a) (Polynomial.scaleRoots p s) = f s ^ Polynomial.natDegree p * Polynomial.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) : Polynomial.eval₂ f (f s * r) (Polynomial.scaleRoots p s) = f s ^ Polynomial.natDegree p * Polynomial.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 : Polynomial.eval₂ f r p = 0) : Polynomial.eval₂ f (f s * r) (Polynomial.scaleRoots p s) = 0

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 : (Polynomial.aeval a) p = 0) : (Polynomial.aeval ((algebraMap R A) r * a)) (Polynomial.scaleRoots p 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 : S} (hr : Polynomial.eval₂ f (f r / f s) p = 0) (hs : s ∈ nonZeroDivisors S) : Polynomial.eval₂ f (f r) (Polynomial.scaleRoots p 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 : R} {s : R} (hr : (Polynomial.aeval ((algebraMap R K) r / (algebraMap R K) s)) p = 0) (hs : s ∈ nonZeroDivisors R) : (Polynomial.aeval ((algebraMap R K) r)) (Polynomial.scaleRoots p s) = 0

@[simp]

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

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

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

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

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

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

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

Alias of Polynomial.scaleRoots_dvd.

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

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

Alias of Polynomial.scaleRoots_dvd_iff.

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

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

Alias of Polynomial.isCoprime_scaleRoots.