Centers for polynomial vector fields of arbitrary degree (Q1006206)

The paper is devoted to the study of the center-focus problem for the origin of planar polynomial systems that in complex coordinates write as \(\dot z= iz+Az^n\bar z^j+B z^k\bar z^l,\) where \(A\) and \(B\) are complex numbers and \(n,j,k\) and \(l\) are non-negative integers. The case \(B=0\) is simple and it turns out that the origin is always a center (which is reversible) unless Re\((A)\neq0\) and \(j=n-1.\) In the general case \(AB\neq0\) and \((n-k)^2+(j-l)^2\neq0\), the authors solve the problem when \(j=n-1\) (or similarly \(l=k-1\)) and give some partial results in the remainder cases.