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Workshop

Singularities of Algebraic Differential Equations

  • Werner Seiler (University of Kassel, Germany)
E1 05 (Leibniz-Saal)

Abstract

We discuss a framework for defining and detecting singularities of arbitrary fully nonlinear systems of ordinary or partial differential equations with polynomial nonlinearities. It combines concepts from differential topology with methods from differential algebra and algebraic geometry. With its help, we provide for the first time a general definition of singularities of partial differential equations and show that it is at least meaningful in the sense that generic points are regular. Our definition is then extended to the notion of a regularity decomposition of a differential equation at a given order and the existence of such decompositions is proven by presenting an algorithm for their effective determination (with an implementation in Maple). Finally, we rigorously define the notion of a regular differential equation (a fundamental concept in the geometric theory of differential equations). We show that our algorithm automatically extracts one provably regular differential equation from each prime component of a given equations and thus provides an effective answer to an old problem in the geometric theory of differential equations.

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Rida Ait El Manssour

Max Planck Institute for Mathematics in the Sciences

Marc Härkönen

Georgia Institute of Technology

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences