Algebraic approximation of germs of real analytic sets

Abstract

Two subanalytic subsets of 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} . .