Abstract

The inverse integrating factor is showed to be a very important tool in the qualitative theory of planar polynomial vector fields X=(P,Q). A function V(x,y) that satisfies the linear partial differential equation div(X/V)=0 is called an inverse integrating factor of X. If X is real and V is an inverse integrating factor of X, then the set V^{-1}(0) contains many information about the skeleton of the phase portrait. Moreover, the existence of an inverse integrating factor V is equivalent to the existence of a first integral H.

If X has a rational first integral H=f/g of degree n, then the remarkable values, i.e. the complex numbers c for which the polynomial f+cg reduces or has degree lower than n, appear to be useful. In particular, we prove that the associated level curves are contained into the set V^{-1}(0).

We present some properties which relate the inverse integrating factor and the remarkable values of a rational (or polynomial) first integral. We also show the importance of the curves f+cg=0 when c is a remarkable value and H=f/g is a rational first integral.