A global Nullstellensatz for ideals of Denjoy-Carleman functions (Q2790922)

Let \({\mathcal C}\) be a quasi-analytic ring of smooth functions in \({\mathbb R}^n\) enjoying the property of resolution of singularities as described by Bierstone and Milman in [Sel. Math., New Ser. 10, No. 1, 1-28 (2004; Zbl 1078.14087)]. The Łojasiewicz radical of an ideal \(I\) in \({\mathcal C}\) is defined as the ideal \(\root{L}\of{I}\) of all functions \(g\in {\mathcal C}\) for which that are a function \(f\in I\) and an integer \(m\geq 1\) such that \(f\geq g^{2m}\) on \({\mathbb{R}}^n\). The saturation of an ideal \(J\) in \({\mathcal C}\) is defined as the ideal \(\widetilde{J}\) of all functions \(g\in {\mathcal C}\) such that for any \(x\in {\mathbb{R}}^n\), we have \(g_x\in J{\mathcal C}_x\), where the subscript \(x\) denotes the germs at \(x\). It is shown that if \(I\) is finitely generated, then the ideal of functions vanishing on the zero variety of \(I\) coincides with \(\widetilde{\root{L}\of{I}}\). When \(I\) is not finitely generated, a similar characterization is given by a variant of the Łojasiewicz radical, computed on all compact subsets of \({\mathbb R}^n\).