Classification and Counting of Planar Quasi-Homogeneous Differential Systems Through Their Weight Vectors

The quasi-homogeneous systems have important properties and they have been studied from various points of view. In this work, we provide the classification of quasi-homogeneous systems on the basis of the weight vector concept, especially in terms of the minimum weight vector, which is proved to be unique for any quasi-homogeneous system. Later we obtain the exact number of different forms of non-homogeneous quasi-homogeneous systems of arbitrary degree, proving a nice relation between this number and Euler’s totient function. Finally, we provide software implementations for some of the above results, and also for the algorithm, recently published by García et al., that generates all the quasi-homogeneous systems.