  Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$.
Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $
C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $
\left[ A_{1},\cdots ,A_{M};C^{m}\right] $, is the ideal of all $f\in
\mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with
each $F_{i}\in C^{m}\left( \mathbb{R}^{n}\right) $.
  In this paper we exhibit an algorithm to compute generators for $\left[
A_{1},\cdots ,A_{M};C^{m}\right] $.
