Approximation of \(C^{\infty}\)-functions without changing their zero-set (Q1120630)

For a \(C^{\infty}\) function \(\phi:\quad M\to {\mathbb{R}}\) (where M is a real algebraic manifold) the following problem is studied. If \(\phi^{- 1}(0)\) is an algebraic subvariety of M, can \(\phi\) be approximated by rational regular functions f such that \(f^{-1}(0)=\phi^{-1}(0)?\) We find that this is possible if and only if there exists a rational regular function \(g:\quad M\to {\mathbb{R}}\) such that \(g^{-1}(0)=\phi^{- 1}(0)\) and g(x)\(\cdot \phi (x)\geq 0\) for any x in \({\mathbb{R}}^ n\). Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus is also studied.