Calculation of singular point quantities at infinity for a type of polynomial differential systems

The center problem at infinity is far to be solved in general. In this paper we develop a procedure to resolve it for a particular type of polynomial differential systems. The problem is solved by writing its concomitant differential equation in the complex coordinates introduced by Yirong Liu and by developing a new method of computation of the so called singular point quantities. This method is based on the transformation of infinity into the elementary origin. Finally, the investigation of center problem for the infinity of a particular family of planar polynomial vector fields of degree 5 is carried out to illustrate the main theoretical results. These involve extensive use of a Computer Algebra System, we have chosen to use Mathematica®.