Tangential approximation of analytic sets

Abstract

Two subanalytic subsets of ${ℝ}^n$ 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 ${ℝ}^n$ 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 $V\left(f\right)$ is the zero set of an analytic map $f$ and if we assume that $V\left(f\right)$ has an isolated singularity, say at the origin $O$, then for any $s\ge 1$ 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 $V\left(f\right)$.