Classification of implication-closed ideals in certain rings of jets

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 $\mathit{\text{closed}}$ 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 $m+n\le 5$. 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 $m+n\le 5$. 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.