Abstract:

To apply the Bendixson-Dulac Theorem to one-parameter families of planar polynomial differential equations one needs to control the sign of certain functions. Many times, these functions are algebraic in $x,y$ and the parameter, and have very high degree. In this talk we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish. It is is based on the computation of the so-called double discriminant and on the study of the singularities (at infinity) of the involved polynomials.

This talk is based on a joint paper with J. D. García-Saldaña and H. Giacomini.