On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity

© 2017, Springer Science+Business Media Dordrecht. Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that for a quadratic polynomial differential system having two pairs of diametrally opposite equilibrium points at infinity, has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.