Local algebraic approximation of semianalytic sets

Algebraic approximation preserving dimension

“…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 : R n → R p we will denote by…”

“…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.…”

“…In the papers [FFW1], [FFW2] and [FFW3] we addressed the question of the existence of an algebraic representative Y in the class of s-equivalence of a given subanalytic set A at a fixed point P . In this case we also say that Y s-approximates A.…”