Ideals in the Ring of m-Jets Induced by Functions that Vanish on Sets

Abstract

Fix positive integers m, n. We work in the ring of m-jets (at the origin) of Cm functions on Rn. We denote this ring Pm. Given a set E in Rn, let I(E) denote the ideal of m-jets (at 0) of C ∞ functions on Rn that vanish on E. Similarly, let Im(E) denote the ideal of m-jets (at 0) of Cm functions on Rn that vanish on E. In [6], the following question is studied. Given an ideal I in the ring of m-jets, how can one tell whether I = Im(E) for some E? In this paper, we study the analogous problem for I(E): given an ideal I, how can we tell whether I = I(E) for some E? We study how the differentiability class impacts the result. In particular, we ask are there ideals that arise as I(E) for some E but do not arise as Im(E) for any E? Similarly, are there ideals that arise as Im(E) for some E but do not arise as I(E) for any E?