A Nullstellensatz for \L ojasiewicz ideals

Abstract

For an ideal of smooth functions that is either {\L}ojasiewicz or weakly {\L}ojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety in terms of the global {\L}ojasiewicz radical and Whitney closure. We also prove that the {\L}ojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to recover in a different way Nullstellensatz results due to Bochnak and Adkins-Leahy and answer positively a modification of the Nullstellensatz conjecture due to Bochnak.