Solutions to a system of equations for $C^m$ functions

Abstract

Fix $m\ge 0$, and let $A=(A_{ij}(x){)}_{1\le i\le N,1\le j\le M}$ be a matrix of semialgebraic functions on ${\mathbb{R}}^n$ or on a compact subset $E\subset {\mathbb{R}}^n$. Given $f=(f_1,\dots ,f_N)\in C^{\infty }({\mathbb{R}}^n,{\mathbb{R}}^N)$, we consider the following system of equations:

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 $F=(F_1,\dots ,F_M)$ if and only if $f=(f_1,\dots ,f_N)$ is annihilated by the linear partial differential operators.