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Local approximation of semialgebraic sets. (English) Zbl 1051.14065
The aim of this paper is to investigate the possibility of locally approximating semialgebraic sets by algebraic ones.
This problem has been considered by L. Bröcker from a different point of view [in: Real algebraic geometry, Lect. Notes Math. 1524, 145–162 (1992; Zbl 0849.14022)]. In this paper the notion of proximity is connected with the Hausdorff distance, namely, two sets \(A\) and \(B\) are said to be \(s\)-equivalent at \(0\) if the Hausdorff distance \(d(A \cap S_r, B \cap S_r)\) tends to \(0\) more rapidly than \(r^s\), where \(S_r\) denotes the sphere of radius \(r\) centered at \(0\).
The main result is the following approximation theorem: Let \(A \subset {\mathbb{R}}^n\) be any closed semialgebraic set of codimension \(\geq 1\) and suppose that \(0\) is a non-isolated point of \(A\). Then for any real number \(s \geq 1\) there exists an algebraic subset \(V \subset {\mathbb{R}}^n\) such that \(A\) and \(V\) are \(s\)-equivalent at \(0\).
This theorem is a generalization of a previous result of the authors [Pac. J. Math. 194, No. 2, 315–323 (2000; Zbl 1036.14027)] saying that any closed semialgebraic subset is \(1\)-equivalent to its tangent cone.

14P10 Semialgebraic sets and related spaces
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