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Gevrey Asymptotic Implicit Function Theorem

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arxiv 2112.08792 v2 pith:ILYVFRZR submitted 2021-12-16 math.CV math-phmath.CAmath.MP

classification math.CVmath-phmath.CAmath.MP
keywords asymptoticgevreytheoremfunctionimplicitboreldefinedimplicitly
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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