A property of ideals of jets of functions vanishing on a set
Charles Fefferman, Ary Shaviv
TL;DR
The paper analyzes which ideals of $\\mathcal{P}_0^m(\\mathbb{R}^n)$ can be realized as $I^m(E)$ for some origin-containing set $E$, by introducing the closure $cl(I)$ of an ideal as the collection of jets implied by $I$ and proving that $I^m(E)$ is always closed. It establishes a robust framework connecting algebraic properties of jets with geometric constraints (allowed/forbidden directions) and develops tools for determining when an ideal implies a jet through negligible and strong-implication notions. A key methodological contribution is the use of semi-algebraic bundles and the stable Glaeser refinement to formulate and solve the closure computation problem, yielding a principled (though computationally heavy) algorithm to obtain $cl(I)$. The results tie into Whitney extension and Nullstellensatz-type perspectives, and the FS2 work extends these ideas to classify closed ideals in several low-dimensional cases, including $m=1$, $n=1$, and $m+n\\leq 5$.
Abstract
For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the 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 an 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 \textit{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 ideal is of the form $I^m(E)$ for some $E$, however we prove in [FS] that all closed ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$.