Gevrey Asymptotic Implicit Function Theorem

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.

Gevrey Asymptotic Existence and Uniqueness Theorem

“…We use the Borel-Laplace method which is briefly reviewed in §A.2. Our proof represents a combination of techniques developed in [Nik20] and [Nik21b] but many of the ideas underpinning all these works originated in [Nik19]. The overall strategy of the proof is as follows.…”