Gevrey Asymptotic Implicit Function Theorem

TL;DR

This paper establishes an implicit function theorem within Gevrey asymptotics, showing that solutions have Gevrey asymptotic expansions and are Borel resummable, with applications to matrix decompositions.

Contribution

It introduces a Gevrey asymptotic implicit function theorem linking formal solutions and Borel resummation, extending classical implicit function results.

Findings

Solutions have Gevrey asymptotic expansions.

Solutions are Borel resummable.

Application to Jordan decomposition of matrices.

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.