On almost Blow-analytic equivalence

TL;DR

This paper investigates the relationship between analytic and Nash equivalence of function germs after desingularization, showing they coincide under certain blow-up procedures using Euclidean and Néron Desingularization techniques.

Contribution

It establishes that Nash equivalence after desingularization matches analytic equivalence for Nash function germs, extending classical approximation results.

Findings

Nash approximation of desingularized analytic functions is possible.

Analytic and Nash equivalences coincide after desingularization.

Uses Euclidean description and Néron Desingularization to prove results.

Abstract

Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of blowings-up along smooth analytic centers. We apply the result to prove that Nash function germs that are analytically equivalent after analytic desingularizations are Nash equivalent after Nash desingularizations. Results are based on a precise Euclidean description of a sequence of blowings-up combined with N\'eron Desingularization.