For a set E ⊂ ℝn that contains the origin we consider Im(E)—the set of all mth degree Taylor approximations (at the origin) of Cm functions on ℝn that vanish on E. This set is a proper ideal in P \(^{m}(\mathbb{R}^{n})\) —the ring of all mth degree Taylor approximations of Cm functions on ℝn. In [FS] we introduced the notion of a closed ideal in P \(^{m}(\mathbb{R}^{n})\) , and proved that any ideal of the form Im(E) is closed. In this paper we classify (up to a natural equivalence relation) all closed ideals in P \(^{m}(\mathbb{R}^{n})\) in all cases in which m + n ≤ 5. We also show that in these cases the converse also holds—all closed proper ideals in P \(^{m}(\mathbb{R}^{n})\) arise as Im(E) when m + n ≤ 5. In addition, we prove that in these cases any ideal of the form Im(E) for some E ⊂ ℝn that contains the origin already arises as Im(V) for some semi-algebraic V ⊂ ℝn that contains the origin. By doing so we prove that a conjecture by N. Zobin holds true in these cases.
