On the non-analyticity locus of an arc-analytic function

TL;DR

This paper proves that the set where an arc-analytic function fails to be analytic is symmetric with respect to real analytic arcs, and shows how certain arc-analytic functions can be made analytic through a sequence of blowings-up.

Contribution

It establishes the arc-symmetry of the non-analyticity locus and demonstrates a method to resolve non-analytic points via blowings-up within that locus.

Findings

Non-analyticity locus is arc-symmetric.

Resolution centers can be chosen inside the non-analyticity locus.

Arc-analytic functions satisfying polynomial equations can be globally resolved.

Abstract

In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. Recall that a function is called arc-analytic if it is real analytic on each real analytic arc. By a result of Bierstone and Milman a big class of arc-analytic function, namely those that satisfy a polynomial equation with real analytic coefficients, can be made analytic by a sequence of global blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity locus.