Blow-analytic equivalence of two variable real analytic function germs

TL;DR

This paper characterizes blow-analytic equivalence for two-variable real analytic function germs using real tree models and minimal resolutions, establishing it as a natural analogue of topological equivalence.

Contribution

It provides complete characterizations of blow-analytic equivalence in two dimensions and shows it can be made cascade, preserving contact orders of arcs.

Findings

Characterization via real tree models and Puiseux pairs

Equivalence can be made cascade and preserves contact orders

In higher dimensions, singular modifications satisfy arc-lifting

Abstract

Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case: in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact orders of real analytic arcs. In the general $n$-dimensional case, we show that a singular real modification satisfies the arc-lifting property.