Algebraic approximation of germs of real analytic sets

Two subanalytic subsets of $\mathbb{R}^n$ are $s$-equivalent at a common point, say $O$, if the Hausdorff distance between their intersections with the sphere centered at $O$ of radius $r$ goes to zero faster than $r^s$. In the present paper we investigate the existence of an algebraic representative in every $s$-equivalence class of subanalytic sets. First we prove that such a result holds for the zero-set $V(f)$ of an analytic map $f$ when the regular points of $f$ are dense in $V(f)$. Moreover we present some results concerning the algebraic approximation of the image of a real analytic map $f$ under the hypothesis that $f^{-1}(O)=\{O\}$.