Note on \(C^\infty\) functions with the zero property (Q1288011)

Let \(M\) be a connected \(C^\infty\) manifold and \(J\) an ideal in the ring \(C^\infty(M)\). \(J\) is said to have \textit{the zero property} if all functions in \(C^\infty(M)\) vanishing on the zeros of \(J\) belong to \(J\). Also, a function \(f\in C^\infty(M)\) is said to have the zero property if the principal ideal \((f)\) has the zero property. \textit{J. Bochnak} [Topology 12, 417-424 (1973; Zbl 0282.58003)] proved that if \(M\) is real analytic and \(f_i:\;M\to\mathbb R,\;f_i\neq 0\;(1\leq i\leq k)\) are real analytic and have the zero property, then \(f=f_1\cdots f_k\) has the zero property iff \(\overline{G(f)}=V(f)\), where \(V(f)= \{f=0\}\) and \(G(f)\) denotes the set of regular points of \(f\) in \(V(f)\). In this paper the author generalizes Bochnak's result by getting rid of the condition of analyticity (on \(M\) and \(f_i\)). Moreover, he adds five equivalent conditions for \(f\) to have the zero property, which are of algebraic or purely topological nature.