Analytic equivalence of normal crossing functions on a real analytic manifold

TL;DR

This paper proves that for real analytic functions with normal crossing singularities, smooth right equivalence implies analytic equivalence, and classifies these functions into a countable set of equivalence classes.

Contribution

It establishes that smooth right equivalence implies analytic equivalence for functions with normal crossing singularities and classifies their equivalence classes as countable or finite.

Findings

Smooth right equivalence implies analytic equivalence for these functions.

The set of equivalence classes is either finite or countably infinite.

Abstract

By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a suitable modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions $C^{\infty}$ right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of equivalence classes is zero or countable.