Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification

Critical points at infinity for autonomous differential systems are defined and used as an essential tool. Rn is mapped onto the unit ball by various mappings and the boundary points of the ball are used to distinguish between different directions at infinity. These mappings are special cases of compactifications. It is proved that the definition of the critical points at infinity is independent of the choice of the mapping to the unit ball.We study the rate of blow up of solutions in autonomous polynomial differential systems of equations via compactification methods. To this end we represent each solution as a quotient of a vector valued function (which is a solution of an associated autonomous system) by a scalar function (which is a solution of a related scalar equation).