Abstract

We deal with polinomial vector fields of the form X = (P,Q) with P,Q in C[x, y]. Maybe the most important question in the Darboux Theory of Integrability is to recognize the existence of an integrating factor formed by invariant algebraic curves (some of these curves could be multiple curves). This is a difficult problem. In this talk, we mainly present (avoided technical details) the following inverse problem: Find all vector fields X having some fixed set of algebraic curves invariants and give an algorithm to construct these vector filds. We also discuss some other kinds of inverse problems.

This is a joint work with C. Christopher, J. Llibre and S. Walcher.