A Nullstellensatz for Łojasiewicz ideals (Q2256082)

The paper concerns a Nullstellensatz for ideals in the ring \(\mathcal{E}(M)\) of real-valued smooth functions on a smooth manifold \(M.\) An ideal \( \mathfrak{a}\subset \mathcal{E}(M)\) is a Łojasiewicz ideal if \(\mathfrak{a} \) is finitely generated and there exists \(f\in \mathfrak{a}\) such that for any compact \(K\subset M\) there exist \(C>0\) and an integer \(m\) such that \( \left| f(x)\right| \geq C\mathrm{d}(x,\mathcal{Z}(\mathfrak{a}))^{m}\) where \( \mathcal{Z}(\mathfrak{a})\) is the zero-set of \(\mathfrak{a}\) and \(\mathrm{d}\) is the distance function. The main result is that if \(\mathfrak{a\subset }\mathcal{E }(M)\) is a Łojasiewicz ideal then for the ideal \(\mathcal{I(Z}(\mathfrak{a} ))\) of functions in \(\mathcal{E}(M)\) vanishing on \(\mathcal{Z}(\mathfrak{a})\) we have \[ \mathcal{I(Z}(\mathfrak{a}))=\overline{^{L}\!\!\!\sqrt{\mathfrak{a}}}, \] where the Łojasiewicz radical \({}^{L}\!\!\!\sqrt{\mathfrak{a}}\) is defined as \(^{L}\!\!\!\sqrt{\mathfrak{a}}:=\{g\in \mathcal{E}(M):\exists _{f\in \mathfrak{a}}\exists _{m\geq 1}f>g^{2m}\) on \(M\}\) and the closure is taken in the compact-open topology. As a corollary the authors obtain known results by \textit{J. Bochnak} [Topology 12, 417--424 (1973; Zbl 0282.58003)], \textit{W. A. Adkins} and \textit{J. V. Leahy} [Duke Math. J. 42, 707--716 (1975; Zbl 0357.46032); Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 60, 90--94 (1976; Zbl 0367.14003)] concerning the case \(\mathfrak{a}\) is generated by analytic functions.