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Mathlib.RingTheory.PowerSeries.Evaluation

Evaluation of power series #

Power series in one indeterminate are the particular case of multivariate power series, for the Unit type of indeterminates. This file provides a simpler syntax.

Let R, S be types, with CommRing R, CommRing S. One assumes that IsTopologicalRing R and IsUniformAddGroup R, and that S is a complete and separated topological R-algebra, with IsLinearTopology S S, which means there is a basis of neighborhoods of 0 consisting of ideals.

Given φ : R →+* S, a : S, and f : MvPowerSeries σ R, PowerSeries.eval₂ f φ a is the evaluation of the power series f at a. It f is (the coercion of) a polynomial, it coincides with the evaluation of that polynomial. Otherwise, it is defined by density from polynomials; its values are irrelevant unless φ is continuous and a is topologically nilpotent (a ^ n tends to 0 when n tends to infinity).

Under Continuous φ and IsTopologicallyNilpotent a, the following lemmas furnish the properties of evaluation:

We refer to the documentation of MvPowerSeries.eval₂ for more details.

@[reducible, inline]
abbrev PowerSeries.HasEval {S : Type u_2} [CommRing S] [TopologicalSpace S] (a : S) :

Points at which evaluation of power series is well behaved

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    theorem PowerSeries.hasEval {S : Type u_2} [CommRing S] [TopologicalSpace S] {a : S} (ha : HasEval a) :
    theorem PowerSeries.HasEval.mono {S : Type u_3} [CommRing S] {a : S} {t u : TopologicalSpace S} (h : t u) (ha : HasEval a) :
    theorem PowerSeries.HasEval.add {S : Type u_2} [CommRing S] [TopologicalSpace S] [ContinuousAdd S] [IsLinearTopology S S] {a b : S} (ha : HasEval a) (hb : HasEval b) :
    HasEval (a + b)
    theorem PowerSeries.HasEval.mul_left {S : Type u_2} [CommRing S] [TopologicalSpace S] [IsLinearTopology S S] (c : S) {x : S} (hx : HasEval x) :
    HasEval (c * x)
    theorem PowerSeries.HasEval.mul_right {S : Type u_2} [CommRing S] [TopologicalSpace S] [IsLinearTopology S S] (c : S) {x : S} (hx : HasEval x) :
    HasEval (x * c)

    The domain of evaluation of MvPowerSeries, as an ideal

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      noncomputable def PowerSeries.eval₂ {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (φ : R →+* S) (a : S) [UniformSpace R] [UniformSpace S] :
      PowerSeries RS

      Evaluation of a power series f at a point a.

      It coincides with the evaluation of f as a polynomial if f is the coercion of a polynomial. Otherwise, it is only relevant if φ is continuous and a is topologically nilpotent.

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        @[simp]
        theorem PowerSeries.eval₂_coe {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (φ : R →+* S) (a : S) [UniformSpace R] [UniformSpace S] (f : Polynomial R) :
        eval₂ φ a f = Polynomial.eval₂ φ a f
        @[simp]
        theorem PowerSeries.eval₂_C {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (φ : R →+* S) (a : S) [UniformSpace R] [UniformSpace S] (r : R) :
        eval₂ φ a ((C R) r) = φ r
        @[simp]
        theorem PowerSeries.eval₂_X {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (φ : R →+* S) (a : S) [UniformSpace R] [UniformSpace S] :
        eval₂ φ a X = a

        The evaluation homomorphism at a on PowerSeries, as a RingHom.

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          theorem PowerSeries.hasSum_eval₂ {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {φ : R →+* S} {a : S} [UniformSpace R] [UniformSpace S] [IsUniformAddGroup R] [IsTopologicalSemiring R] [IsUniformAddGroup S] [T2Space S] [CompleteSpace S] [IsTopologicalRing S] [IsLinearTopology S S] ( : Continuous φ) (ha : IsTopologicallyNilpotent a) (f : PowerSeries R) :
          HasSum (fun (d : ) => φ ((coeff R d) f) * a ^ d) (eval₂ φ a f)
          theorem PowerSeries.eval₂_eq_tsum {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {φ : R →+* S} {a : S} [UniformSpace R] [UniformSpace S] [IsUniformAddGroup R] [IsTopologicalSemiring R] [IsUniformAddGroup S] [T2Space S] [CompleteSpace S] [IsTopologicalRing S] [IsLinearTopology S S] ( : Continuous φ) (ha : IsTopologicallyNilpotent a) (f : PowerSeries R) :
          eval₂ φ a f = ∑' (d : ), φ ((coeff R d) f) * a ^ d
          theorem PowerSeries.eval₂_unique {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {φ : R →+* S} {a : S} [UniformSpace R] [UniformSpace S] [IsUniformAddGroup R] [IsTopologicalSemiring R] [IsUniformAddGroup S] [T2Space S] [CompleteSpace S] [IsTopologicalRing S] [IsLinearTopology S S] ( : Continuous φ) (ha : IsTopologicallyNilpotent a) {ε : PowerSeries RS} ( : Continuous ε) (h : ∀ (p : Polynomial R), ε p = Polynomial.eval₂ φ a p) :
          ε = eval₂ φ a
          theorem PowerSeries.comp_eval₂ {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {φ : R →+* S} {a : S} [UniformSpace R] [UniformSpace S] [IsUniformAddGroup R] [IsTopologicalSemiring R] [IsUniformAddGroup S] [T2Space S] [CompleteSpace S] [IsTopologicalRing S] [IsLinearTopology S S] ( : Continuous φ) (ha : IsTopologicallyNilpotent a) {T : Type u_3} [UniformSpace T] [CompleteSpace T] [T2Space T] [CommRing T] [IsTopologicalRing T] [IsLinearTopology T T] [IsUniformAddGroup T] {ε : S →+* T} ( : Continuous ε) :
          ε eval₂ φ a = eval₂ (ε.comp φ) (ε a)

          For IsTopologicallyNilpotent a, the evaluation homomorphism at a on PowerSeries, as an AlgHom.

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