Planar Semi-quasi Homogeneous Polynomial Differential Systems with a Given Degree

This paper study the planar semi-quasi homogeneous polynomial differential systems (PSQHPDS), which can be regarded as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. As an application of this algorithm, we research the center problem of quadratic and cubic PSQHPDS. The nonexistence of the quadratic center is proved and the canonical form of cubic center is found.