Singularities of Algebraic Differential Equations
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Co-author(s): Markus Lange-Hegermann and Daniel Robertz and Matthias Seiß |
Reference: Advances in Applied Mathematics, 131 (2021) 102266 |
Description: This paper combines 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. The three main results are firstly a proof that even in the
case of partial differential equations regular points are generic. Secondly,
an algorithm for the effective detection of all singularities at a given order
or, more precisely, for the determination of a regularity decomposition is
presented. Finally, a rigorous definition of a regular differential equation, a
notion that is ubiquitous in the geometric theory of differential equations, is
given and it is shown that the above mentioned 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. |
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