Centers of some cubic systems (Q2780922)

This paper is concerned with centers of cubic complex systems. The authors prove that the center variety of the system NEWLINE\[NEWLINEx'=x-a_{10} x^2-a_{01}xy -a_{11}x^2y-a_{02} xy^2,\;y'=-(y-b_{01}y^2-b_{10}xy- b_{11}xy^2-b_{20}x^2y),NEWLINE\]NEWLINE has eight irreducible components.NEWLINENEWLINENEWLINEAlso, they prove that the center variety of the system NEWLINE\[NEWLINEx'=x-a_{10} x^2-a_{01}xy- a_{11}x^2y- a_{20}x^3,\;y'=-(y-b_{01} y^2-b_{10}xy-b_{11} xy^2-b_{02}y^3),NEWLINE\]NEWLINE has seven irreducible components.NEWLINENEWLINENEWLINEBoth results are obtained by computing focal quantities of the considered systems. In both cases, the components admit rational parametrizations.