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

Abstract

Fix $m\geq 0$, and let $A=(A\_{ij}(x))\_{1 \leq i \leq N, 1\leq j \leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^n$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=(f\_1,\ldots,f\_N) \in C^\infty(\mathbb{R}^n, \mathbb{R}^N)$, we consider the following system of equations: $$ \sum\_{j=1}^M A\_{ij} (x) F\_j (x) = f\_i (x) \quad\text{for } i =1,\ldots, N. $$ 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 $F=(F\_1,\ldots,F\_M)$ if and only if $f=(f\_1,\ldots,f\_N)$ is annihilated by the linear partial differential operators.