Formal and finite order equivalences

TL;DR

This paper establishes that formal equivalence of real-analytic germs can be characterized by finite order equivalences, but provides counterexamples in the smooth category, highlighting differences between analytic and smooth cases.

Contribution

It proves that for real-analytic germs, formal equivalence is equivalent to finite order equivalences, and extends this to self-maps and vector fields, while also presenting counterexamples in the smooth setting.

Findings

Formal equivalence coincides with finite order equivalences for real-analytic germs.

Counterexamples show finite order equivalence does not imply formal equivalence in smooth cases.

Techniques developed apply to self-maps and vector fields under conjugation.

Abstract

We show that two families of germs of real-analytic subsets in $C^{n}$ are formally equivalent if and only if they are equivalent of any finite order. We further apply the same technique to obtain analogous statements for equivalences of real-analytic self-maps and vector fields under conjugations. On the other hand, we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent.