Local algebraic approximation of semianalytic sets

Abstract

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

Algebraic approximation preserving dimension

“…The previous theorem allows us to strengthen the following result on approximation preserving dimension which can be found in [FFW3]: Example 4.4. If A = {(x, y, z) ∈ R 3 | z = 0, x ≥ 0, y ≥ 0} and s ≥ 1, the approximation technique described in the proof of Theorem 4.1 yields a surface defined by (z 2 −x m ) 2 −y p = 0 for suitable odd integers m and p; the shape of such a surface is represented in Figure 1…”

“…In this paper we prove in Theorem 4.1 that any semialgebraic set of codimension ≥ 1 is s-equivalent to an algebraic one of the same dimension. Using the mentioned result of [FFW3], we obtain (Corollary 4.3) that any semianalytic set of codimension ≥ 1 can be s-approximated by an algebraic one preserving the local dimension. The proof of Theorem 4.1 works provided that the semialgebraic set is described by means of a suitable presentation, as in the previous example.…”

“…Another essential tool will be the following version of Lojasiewicz' inequality, proved in [FFW3]; henceforth for any map f :…”

“…Proof. Arguing as in [FFW3,Lemma 3.2] in the semialgebraic setting, there exist semialgebraic sets Γ 1 , . .…”

“…In [FFW3] we proved that, for any s ≥ 1, any closed semianalytic subset A ⊆ R n is sequivalent to a semialgebraic set Y ⊂ R n having the same local dimension as A. However the arguments used in the proof of this latter result do not guarantee that, even if A is analytic, it can be approximated by means of an algebraic one of the same dimension.…”