A note on a separation problem
The subject of this note is the following problem: Let \(A_ 0\), \(Z_ 0\) be semianalytic closed germs in \({\mathbb{R}}^ n_ 0\) with \(A_ 0\cap Z_ 0=\{0\}\). Under what conditions does there exist an analytic function germ h, positive on \(Z_ 0\backslash\{0\}\) and negative on \(A_ 0\backslash\{0\}?\) It is proven that a sufficient condition is dim \(A_ 0=1\), and in that case h can be chosen polynomial. This is applied to solve in low dimensions some open problems on semialgebraic sets. For instance, it is shown: (I) Let U be a 1- or 2-dimensional open semialgebraic set of a real algebraic variety. If U is basic, so is \(\bar U\). - (II) Let V be a real algebraic curve and U an open semialgebraic subset of V. Then \(U=\{h>0\}, \bar U=\{h\geq 0\}\) for some polynomial \(h\in {\mathbb{R}}[V]\). (III) Let V be a real algebraic surface and \(V_ c\) its maximum dimension locus. Then \(V_ c=\{h\geq 0\}\) for a polynomial \(h\in {\mathbb{R}}[V]\).
scientific article; zbMATH DE number 3871541