  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.
