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 $A_1,\cdots ,A_M\in \mathcal{R}$. 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 $f\in \mathcal{R}$ 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 $\left[A_1,\cdots ,A_M;C^m\right]$.

47 pages, see also the related article "Solutions to a System of Equations for $C^m$ Functions"