Theory of monic polynomials

We define integralNormalization, which relate arbitrary polynomials to monic ones.

noncomputable def Polynomial.integralNormalization {R : Type u} [Semiring R] (f : Polynomial R) : Polynomial R

If f : R[X] is a nonzero polynomial with root z, integralNormalization f is a monic polynomial with root leadingCoeff f * z.

Moreover, integralNormalization 0 = 0.

Equations

• f.integralNormalization = f.support.sum fun (i : ℕ) => (Polynomial.monomial i) (if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i))

@[simp]

theorem Polynomial.integralNormalization_zero {R : Type u} [Semiring R] : Polynomial.integralNormalization 0 = 0

theorem Polynomial.integralNormalization_coeff {R : Type u} [Semiring R] {f : Polynomial R} {i : ℕ} : f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)

theorem Polynomial.integralNormalization_support {R : Type u} [Semiring R] {f : Polynomial R} : f.integralNormalization.support ⊆ f.support

theorem Polynomial.integralNormalization_coeff_degree {R : Type u} [Semiring R] {f : Polynomial R} {i : ℕ} (hi : f.degree = ↑i) : f.integralNormalization.coeff i = 1

theorem Polynomial.integralNormalization_coeff_natDegree {R : Type u} [Semiring R] {f : Polynomial R} (hf : f ≠ 0) : f.integralNormalization.coeff f.natDegree = 1

theorem Polynomial.integralNormalization_coeff_ne_degree {R : Type u} [Semiring R] {f : Polynomial R} {i : ℕ} (hi : f.degree ≠ ↑i) : f.integralNormalization.coeff i = f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)

theorem Polynomial.integralNormalization_coeff_ne_natDegree {R : Type u} [Semiring R] {f : Polynomial R} {i : ℕ} (hi : i ≠ f.natDegree) : f.integralNormalization.coeff i = f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)

theorem Polynomial.monic_integralNormalization {R : Type u} [Semiring R] {f : Polynomial R} (hf : f ≠ 0) : f.integralNormalization.Monic

@[simp]

theorem Polynomial.support_integralNormalization {R : Type u} [Ring R] [IsDomain R] {f : Polynomial R} : f.integralNormalization.support = f.support

theorem Polynomial.integralNormalization_eval₂_eq_zero {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommSemiring S] {p : Polynomial R} (f : R →+* S) {z : S} (hz : Polynomial.eval₂ f z p = 0) (inj : ∀ (x : R), f x = 0 → x = 0) : Polynomial.eval₂ f (z * f p.leadingCoeff) p.integralNormalization = 0

theorem Polynomial.integralNormalization_aeval_eq_zero {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommSemiring S] [Algebra R S] {f : Polynomial R} {z : S} (hz : (Polynomial.aeval z) f = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0) : (Polynomial.aeval (z * (algebraMap R S) f.leadingCoeff)) f.integralNormalization = 0