Singularities of Algebraic Differential Equations

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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