Polynomials and limits

In this file we prove the following lemmas.

• Polynomial.continuous_eval₂: Polynomial.eval₂ defines a continuous function.
• Polynomial.continuous_aeval: Polynomial.aeval defines a continuous function; we also prove convenience lemmas Polynomial.continuousAt_aeval, Polynomial.continuousWithinAt_aeval, Polynomial.continuousOn_aeval.
• Polynomial.continuous: Polynomial.eval defines a continuous functions; we also prove convenience lemmas Polynomial.continuousAt, Polynomial.continuousWithinAt, Polynomial.continuousOn.
• Polynomial.tendsto_norm_atTop: λ x, ‖Polynomial.eval (z x) p‖ tends to infinity provided that fun x ↦ ‖z x‖ tends to infinity and 0 < degree p;
• Polynomial.tendsto_abv_eval₂_atTop, Polynomial.tendsto_abv_atTop, Polynomial.tendsto_abv_aeval_atTop: a few versions of the previous statement for IsAbsoluteValue abv instead of norm.

Tags

Polynomial, continuity

theorem Polynomial.continuous_eval₂ {R : Type u_1} {S : Type u_2} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] [Semiring S] (p : Polynomial S) (f : S →+* R) : Continuous fun x => Polynomial.eval₂ f x p

theorem Polynomial.continuous {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) : Continuous fun x => Polynomial.eval x p

theorem Polynomial.continuousAt {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) {a : R} : ContinuousAt (fun x => Polynomial.eval x p) a

theorem Polynomial.continuousWithinAt {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) {s : Set R} {a : R} : ContinuousWithinAt (fun x => Polynomial.eval x p) s a

theorem Polynomial.continuousOn {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) {s : Set R} : ContinuousOn (fun x => Polynomial.eval x p) s

theorem Polynomial.continuous_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) : Continuous fun x => ↑(Polynomial.aeval x) p

theorem Polynomial.continuousAt_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) {a : A} : ContinuousAt (fun x => ↑(Polynomial.aeval x) p) a

theorem Polynomial.continuousWithinAt_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) {s : Set A} {a : A} : ContinuousWithinAt (fun x => ↑(Polynomial.aeval x) p) s a

theorem Polynomial.continuousOn_aeval {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) {s : Set A} : ContinuousOn (fun x => ↑(Polynomial.aeval x) p) s

theorem Polynomial.tendsto_abv_eval₂_atTop {R : Type u_1} {S : Type u_2} {k : Type u_3} {α : Type u_4} [Semiring R] [Ring S] [LinearOrderedField k] (f : R →+* S) (abv : S → k) [IsAbsoluteValue abv] (p : Polynomial R) (hd : 0 < Polynomial.degree p) (hf : ↑f (Polynomial.leadingCoeff p) ≠ 0) {l : Filter α} {z : α → S} (hz : Filter.Tendsto (abv ∘ z) l Filter.atTop) : Filter.Tendsto (fun x => abv (Polynomial.eval₂ f (z x) p)) l Filter.atTop

theorem Polynomial.tendsto_abv_atTop {R : Type u_1} {k : Type u_2} {α : Type u_3} [Ring R] [LinearOrderedField k] (abv : R → k) [IsAbsoluteValue abv] (p : Polynomial R) (h : 0 < Polynomial.degree p) {l : Filter α} {z : α → R} (hz : Filter.Tendsto (abv ∘ z) l Filter.atTop) : Filter.Tendsto (fun x => abv (Polynomial.eval (z x) p)) l Filter.atTop

theorem Polynomial.tendsto_abv_aeval_atTop {R : Type u_1} {A : Type u_2} {k : Type u_3} {α : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] [LinearOrderedField k] (abv : A → k) [IsAbsoluteValue abv] (p : Polynomial R) (hd : 0 < Polynomial.degree p) (h₀ : ↑(algebraMap R A) (Polynomial.leadingCoeff p) ≠ 0) {l : Filter α} {z : α → A} (hz : Filter.Tendsto (abv ∘ z) l Filter.atTop) : Filter.Tendsto (fun x => abv (↑(Polynomial.aeval (z x)) p)) l Filter.atTop

theorem Polynomial.tendsto_norm_atTop {α : Type u_1} {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] (p : Polynomial R) (h : 0 < Polynomial.degree p) {l : Filter α} {z : α → R} (hz : Filter.Tendsto (fun x => ‖z x‖) l Filter.atTop) : Filter.Tendsto (fun x => ‖Polynomial.eval (z x) p‖) l Filter.atTop

theorem Polynomial.exists_forall_norm_le {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] [ProperSpace R] (p : Polynomial R) : ∃ x, ∀ (y : R), ‖Polynomial.eval x p‖ ≤ ‖Polynomial.eval y p‖

theorem Polynomial.eq_one_of_roots_le {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {p : Polynomial F} {f : F →+* K} {B : ℝ} (hB : B < 0) (h1 : Polynomial.Monic p) (h2 : Polynomial.Splits f p) (h3 : ∀ (z : K), z ∈ Polynomial.roots (Polynomial.map f p) → ‖z‖ ≤ B) : p = 1

theorem Polynomial.coeff_le_of_roots_le {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {p : Polynomial F} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : Polynomial.Monic p) (h2 : Polynomial.Splits f p) (h3 : ∀ (z : K), z ∈ Polynomial.roots (Polynomial.map f p) → ‖z‖ ≤ B) : ‖Polynomial.coeff (Polynomial.map f p) i‖ ≤ B ^ (Polynomial.natDegree p - i) * ↑(Nat.choose (Polynomial.natDegree p) i)

theorem Polynomial.coeff_bdd_of_roots_le {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {B : ℝ} {d : ℕ} (f : F →+* K) {p : Polynomial F} (h1 : Polynomial.Monic p) (h2 : Polynomial.Splits f p) (h3 : Polynomial.natDegree p ≤ d) (h4 : ∀ (z : K), z ∈ Polynomial.roots (Polynomial.map f p) → ‖z‖ ≤ B) (i : ℕ) : ‖Polynomial.coeff (Polynomial.map f p) i‖ ≤ max B 1 ^ d * ↑(Nat.choose d (d / 2))

The coefficients of the monic polynomials of bounded degree with bounded roots are uniformly bounded.