On the number of algebraically independent Poincaré-Liapunov constants

In this paper an upper bound for the number of algebraically independent Poincaré-Liapunov constants in a certain basis for planar polynomial differential systems is given. Finally, it is conjectured that an upper bound for the number of functionally independent Poincaré-Liapunov quantities would be m + 3m− 7 where m is the degree of the polynomial differential system. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and in the determination of the center cases are also discussed.