Gevrey Asymptotic Implicit Function Theorem

Nikita Nikolaev*

*Corresponding author for this work

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Abstract

We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of the corresponding implicitly defined formal power series solution. The main theorem can therefore be rephrased as an Implicit Function Theorem for Borel summable power series. As an application, we give a diagonal or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.
Original languageEnglish
JournalL’Enseignement Mathématique
Early online date11 Jan 2024
DOIs
Publication statusE-pub ahead of print - 11 Jan 2024

Keywords

  • exact perturbation theory
  • singular perturbation theory
  • Borel summation
  • Borel-Laplace theory
  • asymptotic analysis
  • Gevrey asymptotics
  • resurgence
  • exact WKB analysis
  • complex analysis
  • Ordinary differential equations
  • WKB

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