A Nullstellensatz for Łojasiewicz ideals.

For an ideal of smooth functions a that is either łojasiewicz or weakly łojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety I(Z(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. © European Mathematical Society.