Solutions to a System of Equations for $C^m$ Functions

Abstract

Fix $m\ge 0$, and let $A={\left(A_{ij}\left(x\right)\right)}_{1\le i\le N,1\le j\le M}$ be a matrix of semialgebraic functions on ${\mathbb{R}}^n$ or on a compact subset $E\subset {\mathbb{R}}^n$. Given $f=\left(f_1,\cdots ,f_N\right)\in C^{\infty }\left({\mathbb{R}}^n,{\mathbb{R}}^N\right)$, we consider the following system of equations \begin{equation} \sum_{j=1}^{M}A_{ij}\left( x\right) F_{j}\left( x\right) =f_{i}\left( x\right) \text{ }\left( i=1,\cdots ,N\right) \text{.} \end{equation} In this paper, we give algorithms for computing a finite list of linear partial differential operators such that $AF=f$ admits a $C^m({\mathbb{R}}^n,{\mathbb{R}}^M)$ solution $F=(F_1,\cdots ,F_M)$ if and only if $f=(f_1,\cdots ,f_N)$ is annihilated by the linear partial differential operators.

48 pages, see also the related paper "Generators for the $C^m$-closures of Ideals"