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Convergent commutative power series

Lax825748.ConvergentSeries · concepts/Lax825748/ConvergentSeries.lean · lax-825748

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    Natural Language Statement

    Definition

    A real power series in dd commuting variables is convergent if its coefficients admit a geometric bound ∣fI∣≤αC∣I∣|f_I|\leq\alpha C^{|I|} with α,C>0\alpha,C>0. For finitely many variables this is equivalent to absolute convergence on a neighborhood of the origin. Dimension zero is included.

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    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Lax825748.FormalSeries
    2
    3/-!
    4---
    5title: Convergent commutative power series
    6type: definition
    7---
    8A real power series in dd commuting variables is convergent if its
    9coefficients admit a geometric bound
    10∣fI∣≤αC∣I∣|f_I|\leq\alpha C^{|I|} with α,C>0\alpha,C>0.
    11For finitely many variables this is equivalent to absolute convergence
    12on a neighborhood of the origin. Dimension zero is included.
    13-/
    14
    15noncomputable section
    16
    17namespace Lax825748.ConvergentSeries
    18
    19open Lax825748.FormalSeries Lax825748.Monomials
    20
    21def HasGeometricBound {d : ℕ} (f : StateSeries d) (α C : ℝ) : Prop :=
    22 ∀ I : Fin d →₀ ℕ, |MvPowerSeries.coeff I f| ≤ α * C ^ totalDegree I
    23
    24def Convergent {d : ℕ} (f : StateSeries d) : Prop :=
    25 ∃ α : ℝ, 0 < α ∧ ∃ C : ℝ, 0 < C ∧ HasGeometricBound f α C
    26
    27end Lax825748.ConvergentSeries
    28
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