Complexity of global semianalytic sets in a real analytic manifold of dimension 2 (Q2717083)

Let \(X \subset\mathbb{R}^n\), be a paracompact, real, connected, analytic manifold of dimension \(2\). A subset of \(X\), of the form \(\{x\in X\mid f(x)>0\}\) (resp. \(\{x\in X|f(x)\geq 0\}\)), \(f\) a global analytic function on \(X\), is called principal open (resp. closed). A finite intersection of principal open (resp. closed) sets is called basic open (resp. closed). NEWLINENEWLINENEWLINEOne has the following invariants: \(s(X)=\inf\{s\in N \mid\) any basic open set can be written using \(s\) global analytic functions\}, and \(t(X)=\inf\{ r\in N\mid\) any finite union of basic open sets can be written as a union of \(r\) basic open sets\}. Similarly one defines \(\bar s(X)\) and \(\bar t(X)\) for basic closed subsets. NEWLINENEWLINENEWLINEThe authors prove that likewise in the semialgebraic case, one has \(s(X)=2\). They also manage to compute the other invariants \(t(X)= \bar s(X)=3\) and \(\bar t(X)=2,\) in this case when \(X\) is a global non-compact analytic set, case when not too much is known. Among other things they show that the Hörmander-Lojasiewicz inequality holds, and they prove the Finiteness Theorem, namely any open (closed) subset of \(X\) is a finite union of basic open (closed) sets.