  Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq
N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or
on a compact subset $E \subset \mathbb{R}^n$. Given $f=\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.
