We combine algebraic and geometric approaches to general systems of algebraic
ordinary or partial differential equations to provide a unified framework for
the definition and detection of singularities of a given system at a fixed
order. 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
notion that is ubiquitous in the geometric theory of differential equations,
and show that our algorithm extracts from each prime component a regular
differential equation. Our main algorithmic tools are on the one hand the
algebraic resp. differential Thomas decomposition and on the other hand the
Vessiot theory of differential equations.Comment: 45 pages, 5 figure
