Local algebraic approximation of semianalytic sets

TL;DR

This paper proves that any closed semianalytic set can be locally approximated by a semialgebraic set of the same dimension to any desired order, bridging the gap between semianalytic and algebraic sets.

Contribution

It establishes that each s-equivalence class of a semianalytic set contains a semialgebraic representative of the same dimension, enabling precise local algebraic approximation.

Findings

Semianalytic sets can be approximated by semialgebraic sets of any order s.

Such approximations preserve the dimension of the original set.

Results extend to algebraic approximation for sets of codimension at least 1.

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 of order greater than s when r tends to 0. In this paper we prove that every s-equivalence class of a closed semianalytic set contains a semialgebraic representative of the same dimension. In other words any semianalytic set can be locally approximated of any order s by means of a semialgebraic set and hence, by previous results, also by means of an algebraic one (so long as the semianalytic set has codimension at least 1).