  We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree
$m>1$. To each algebraic invariant curve of such a field we associate a compact
Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The
asymptotic estimate of the degree of an arbitrary algebraic invariant curve is
found. In the smooth case this estimate was already found by D. Cerveau and A.
Lins Neto [Ann. Inst. Fourier Grenoble 41, 883-903] in a different way.
