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.

中文翻译：

---

$ C ^ m $函数方程组的解

固定$ m \ geq 0 $，然后让$ A =（A_ {ij}（x））_ {1 \ leq i \ leq N，1 \ leq j \ leq M} $是$ \上的半代数函数矩阵mathbb {R} ^ n $或在紧凑子集$ E \ subset \ mathbb {R} ^ n $上。给定$ f =（f_1，\ ldots，f_N）\ in C ^ \ infty（\ mathbb {R} ^ n，\ mathbb {R} ^ N）$，我们考虑以下方程组：\ [\ sum_ { j = 1} ^ MA_ {ij}（x）F_j（x）= f_i（x）\ quad \ text {for} i = 1，\ ldots，N。\]在本文中，我们给出了计算a线性偏微分算子的有限列表，例如$ AF = f $允许$ C ^ m（\ mathbb {R} ^ n，\ mathbb {R} ^ M）$解$ F =（F_1，\ ldots，F_M）当且仅当$ f =（f_1，\ ldots，f_N）$被线性偏微分算子消除时，才使用$。