Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree (Q1918342)

The existence of centers at the origin of the autonomous polynomial system \(\dot x= y+ P(x, y)\), \(\dot y= - x+ Q(x, y)\), where \(P(x, y)\) and \(Q(x, y)\) are polynomials of degree \(n\) but without linear terms, is investigated. New sufficient conditions for a center at the origin are obtained for the case \(n= 3\), and, by induction, the results are generalized to polynomials of arbitrary degree. Extensive use is made of Mathematica for the symbolic computation. Specific applications or examples are included.