A Nullstellensatz for \L ojasiewicz ideals

TL;DR

This paper characterizes the ideal of functions vanishing on the variety of Lojasiewicz ideals using the global Lojasiewicz radical and Whitney closure, providing new insights into Nullstellensatz results for smooth functions.

Contribution

It offers a complete characterization of vanishing ideals for Lojasiewicz and weakly Lojasiewicz ideals, connecting radicals, Whitney closure, and Nullstellensatz.

Findings

Characterization of vanishing ideals via Lojasiewicz radical and Whitney closure

Proof that the Lojasiewicz radical's saturation equals its Whitney closure

Recovery and extension of Nullstellensatz results for smooth functions

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.