Tangential approximation of analytic sets

Abstract

Two subanalytic subsets of ℝn are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes to order >s as r tends to 0. We strengthen this notion in the case of real subanalytic subsets of ℝn with isolated singular points, introducing the notion of tangential s-equivalence at a common singular point, which considers also the distance between the tangent planes to the sets near the point. We prove that, if V(f) is the zero set of an analytic map f and if we assume that V(f) has an isolated singularity, say at the origin O, then for any s≥1 the truncation of the Taylor series of f of sufficiently high order defines an algebraic set with isolated singularity at O which is tangentially s-equivalent to V(f).