On the number of linear particular integrals of algebraic differential equations (Q2515261)

Consider the planar autonomous system \[ {dx\over dt}= P(x,y),\quad {dy\over dt}= Q(x,y).\tag{\(*\)} \] Under the assumption that \(P\) and \(Q\) are coprime polynomials with \(\max(\deg P,\deg Q)=n\), the authors prove that \((*)\) has at most \(3n-1\) distinct linear particular integrals and that this bound is sharp.