There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts of the theory algorithmic. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notoriously difficult notion, ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

存在一个完善的常微分方程奇点的微分拓扑理论。主要研究低阶标量方程。我们建议将关键概念扩展到常微分方程或偏微分方程的任意系统。此外，我们展示了这种几何理论与（微分）代数工具的结合如何使我们能够使部分理论算法化。我们的三个主要结果首先证明，即使在偏微分方程的情况下，正则点也是通用的。其次，我们提出了一种算法，用于以给定的顺序有效检测所有奇点，或者更准确地说，用于确定规律性分解。最后，我们给出了正则微分方程的严格定义，这是一个众所周知的困难概念，在微分方程的几何理论中无处不在，并表明我们的算法从每个素数分量中提取了一个正则微分方程。我们的主要工具一方面是代数。微分 Thomas 分解，另一方面是微分方程的 Vessiot 理论。