On the centers of the weight-homogeneous polynomial vector fields on the plane (Q837612)

The work deals with classification of centers for a family of polynomial differential systems, the so-called weight-homogeneous planar polynomial differential systems of the form \[ \dot{x}=P(x,y), \dot{y}=Q(x,y), \] where \(P, Q\) are polynomials with real coefficients. This system is called weight-homogeneous if it satisfies the conditions: \[ P(\lambda^{a}x,\lambda^{b}y)=\lambda^{a-1+d}P(x,y), \] \[ Q(\lambda^{a}x,\lambda^{b}y)=\lambda^{b-1+d}Q(x,y) \] for some natural numbers \(a,b,d\) and a positive real number \(\lambda.\) A complete characterisation of centers is given in the cases of weight degree 1,2,3,4 and coprime polynomials \(P, Q\).