On the Nullstellensätze for Stein spaces and \(C\)-analytic sets (Q2790612)

Let \(X\) be either a Stein space or a \(C\)-analytic (in the sense of Cartan) subset of \(\mathbb R^n\), and let \(I\) be an ideal in the ring \(\mathcal O(X)\) of analytic functions on \(X\). The goal of the paper is to describe the ideal \(\mathcal I(V)\) of all elements of \(\mathcal O(X)\) vanishing on the zero variety \(V\) of \(I\). In the complex case of Stein spaces, it is shown that \(\mathcal I(V)\) is the closure (in the Fréchet space topology of \(\mathcal O(X)\)) of the radical of \(I\). This statement extends \textit{O. Forster}'s Nullstellensatz for closed ideals [Math. Ann. 154, 307--329 (1964; Zbl 0135.12602)]. In the real case of \(C\)-analytic subsets of \(\mathbb R^n\), a similar description is obtained as follows. The Łojasiewicz radical of \(I\) is defined as the ideal \(\root{L}\of{I}\) of all functions \(g\in {\mathcal O}(X)\) for which there are a function \(f\in I\) and an integer \(m\geq 1\) such that \(f- g^{2m}\geq 0\). The saturation of an ideal \(J\) in \({\mathcal O}(X)\) is defined as the ideal \(\widetilde{J}\) of all functions \(g\in {\mathcal O}(X)\) such that for any \(x\in X\), we have \(g_x\in J{\mathcal O}_{X,x}\), where the subscript \(x\) denotes the germs at \(x\). It is then shown that \({\mathcal I}(V)=\widetilde{\root{L}\of{I}}\). A number of other properties are discussed. For instance, under suitable additional assumptions on \(V\) in terms of Hilbert's 17th problem, it is shown that the Łojasiewicz radical can be replaced by other notions of radical whose definitions involve sums of squares, namely the classical real radical of \(I\), or its so-called real-analytic radical.