A property of ideals of jets of functions vanishing on a set

Abstract

For a set $E\subset\mathbb{R}^n$ that contains the origin, we consider $I^m(E)$ – the set set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish on $E$. This set is a proper ideal in $\mathcal{P}^m(\mathbb{R}^n)$ – the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. Which ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ for some $E$? In this paper we introduce the notion of a closed ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and prove that any ideal of the form $I^m(E)$ is closed. We do not know whether in general any closed proper ideal is of the form $I^m(E)$ for some $E$, however we prove in a subsequent paper that all closed proper ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$.