Stability index of closed semianalytic set germs (Q1297949)

Let \(X_0\) be an analytic set germ at \(0\in \mathbb R^n\). Its stability index \(\bar s(X_0)\) is defined as the minimum of all \(s\in \mathbb Z\) such that any basic closed semianalytic set germ \(S_0\subset X_0\) can be written as \(S_0=\{g_1\geq 0, \dots, g_s\geq 0\}\) by using \(s\) analytic function germs \(g_1, \dots, g_s\) at \(X_0\). The index \(\bar s(d)\) is then defined as the maximum of \(\bar s(X_0)\) for all \(d\)-dimensional analytic set germ \(X_0\). In the present paper it is shown that \(\bar s(X_0)=2\) for any \(2\)-dimensional normal analytic set germ and that \(\bar s(d)=d(d+1)/2\) for \(d>2\).