On the number of algebraic invariant curves of polynomial vector fields (Q2387959)

Consider the autonomous system \[ \frac{dx}{dt} = P(x,y), \;\frac{dy}{dt} = Q(x,y),\tag{1} \] in the plane, where \(P\) and \(Q\) are coprime polynomials and \(\max (\deg P, \deg Q)=n.\) The author proves that if the set of trajectories of (1) contains only finitely many pairwise disjoint algebraic curves irreducible over the field of complex numbers, then the number \(s\) of these curves satisfies \[ s \leq \frac{n^2 + n +2}{2}. \] This estimate is sharp for \(n=2\).