Singularities of Algebraic Differential Equations

Abstract

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.

A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations

“…In [33], the first two authors developed together with collaborators a novel framework for the analysis of algebraic differential equations, i. e. differential equations (and inequations) described by differential polynomials, which combines ideas and techniques from differential algebra, differential geometry and algebraic geometry. 2 A key role in this new effective approach is played by the Thomas decomposition which exists in an algebraic version for algebraic systems and in a differential version for differential systems.…”

“…Unfortunately, the algorithms behind the Thomas decomposition require that the underlying field is algebraically closed. Hence, it is always assumed in [33] that a complex differential equation is treated. However, most differential equations appearing in applications are real.…”

“…However, most differential equations appearing in applications are real. The main goal of this work is to adapt the framework of [33] to real ordinary differential equations.…”

“…The approach in [33] consists of a differential and an algebraic step. For the prepatory differential step, one may continue to use basic differential algebraic algorithms (for example the differential Thomas decomposition).…”

“…Over the reals, also the treatment of inequalities like positivity conditions is possible which is important for many applications e. g. in biology and chemistry. We will extend the approach from [33] by generalising the notion of an algebraic differential equation used in [33] to semialgebraic differential equations, which allow for arbitrary inequalities. As a further improvement, we will make stronger use of the fact that the detection of singularities represents essentially a linear problem.…”