Classification of implication-closed ideals in certain rings of jets

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 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 $\mathbb{R}^n$. In [FS] we introduced the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and proved that any ideal of the form $I^m(E)$ is closed. In this paper we classify (up to a natural equivalence relation) all closed ideals in $\mathcal{P}^m(\mathbb{R}^n)$ in all cases in which $m+n\leq5$. We also show that in these cases the converse also holds -- all closed proper ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$. In addition, we prove that in these cases any ideal of the form $I^m(E)$ for some $E\subset\mathbb{R}^n$ that contains the origin already arises as $I^m(V)$ for some semi-algebraic $V\subset\mathbb{R}^n$ that contains the origin. By doing so we prove that a conjecture by N. Zobin holds true in these cases.