A Nullstellensatz for Łojasiewicz ideals

Abstract

For an ideal of smooth functions $\mathfrak{a}$ that is either Łojasiewicz or weakly Łojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety $\mathcal{I}(\mathcal{Z}(\mathfrak{a}))$ in terms of the global Łojasiewicz radical and Whitney closure. We also prove that the Łojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to revisit Nullstellensatz results due to Bochnak and Adkins–Leahy and to resolve positively a modification of the Nullstellensatz conjecture due to Bochnak.