Abstract

Given a planar polynomial vector field, there is the classical problem (going back to Darboux and Poincare ) to determine an integrating factor of Darboux type, or to verify that no such factor exist. In this talk we present the inverse problem: Give a general characterization of vector fields with prescribed integrating factors in the affine nondegenerate geometric scenario. The main result is that, modulo a subspace of ‘known’ vector fields admitting a prescribed integrating factor, only a finite dimensional problem remains. To obtain this result, we will employ a quite simple reduction principle, which leads to a system of meromorphic linear differential equations for functions of one variable, and more precisely to polynomialsolutions of this system. For several classes of examples we will prove that the dimension of a certain quotient space is zero, but we will also exhibit cases when the dimension is positive.

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