Two subanalytic subsets of R^n are called s-equivalent at a common point P \ud if the Hausdorff distance between their intersections with the sphere centered \ud at P of radius r vanishes to order >s when r tends to 0. In this paper \ud we prove that every s-equivalence class of a closed \ud semianalytic set contains a semialgebraic \ud representative of the same dimension. In other words any semianalytic set can be \ud locally approximated to any order s by means of a semialgebraic set and hence, \ud by previous results, also by means of an algebraic one

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 sequivalence 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}.

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We prove that each semialgebraic subset of R^n of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving dimension holds also for semianalytic sets