A Nullstellensatz for ideals of $C^\infty$ functions in dimension 2

Abstract

Suppose that an ideal $J$ of $C^\infty$ functions on an open subset of $\mathbf{R}^2$ is a Łojasiewicz ideal. We describe the set of $C^\infty$ functions vanishing on the zeros of $J$ explicitly using $J$ in an open neighborhood of each point in zeros of $J$, it can be obtained by taking real radical and closure starting from $J$ repeatedly for a finite number of times. This gives an another affirmative answer to Bochnak's conjecture in dimension 2, which is first done by Risler.