Rational jet dependence of formal equivalences between real-analytic hypersurfaces in \C^2

TL;DR

This paper proves that formal equivalences between certain real-analytic hypersurfaces in c2^2 are uniquely determined by a finite jet, leading to insights into their local transformation groups.

Contribution

It establishes that formal equivalences are parametrized by finite jets, providing a new understanding of their structure and uniqueness.

Findings

Formal equivalences are determined by finite jets.

The order of jet needed depends only on the hypersurface.

Application to local formal transformation groups.

Abstract

Let (M,p) and (M',p') be the germs of real-analytic 1-infinite type hypersurfaces in \C^2. We prove that any formal equivalence sending (M,p) into (M',p') is formally parametrized (and hence uniquely determined by) its jet at p of a predetermined order depending only on (M,p). As an application, we use this to examine the local formal transformation groups of such hypersurfaces.