Abstract

Polynomial ordinary differential equations in dimension two continue to be in the focus of interest. Invariant algebraic curves play a special role since they are naturally associated with certain classes of integrating factors or, stated differently, with certain classes of nonlinear symmetries. Understanding such invariant curves and integrating factors is crucial for the investigation of qualitative properties of such equations.
Determining the invariant algebraic curves of a given polynomial system (or proving that no such curves exist) is a very hard problem. The "inverse problem" to determine all plane vector fields which admit a given collection of algebraic curves as invariant set is also of interest. In the main part of the talk we will show that, and how, it can be solved using classical results from Commutative Algebra.