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 m2 + 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.