Generators for the $C^m$-closures of ideals

Abstract

Let $\mathcal{R}$ denote the ring of real polynomials on ${\mathbb{R}}^n$. Fix $m\ge 0$, and let $A_1,\dots ,A_M\in \mathcal{R}$. The $C^m$-closure of $(A_1,\dots ,A_M)$, denoted here by $[A_1,\dots ,A_M;C^m]$, is the ideal of all $f\in \mathcal{R}$ expressible in the form $f=F_1{A}_1+\cdots +F_M{A}_M$ with each $F_i\in C^m({\mathbb{R}}^n)$. In this paper we exhibit an algorithm for computing generators for $[A_1,\dots ,A_M;C^m]$.