Vector fields with simply connected trajectories transverse to a polynomial.

We classify the polynomial vector fields X on C 2 which have simply-connected trajectories and satisfy d P ( X ) = 1 for P in C [ x , y ] . In particular, we obtain that such vector fields are complete and have all its trajectories proper and of type C . Finally, we apply this result to classify surjective C -derivations of C [ x , y ] , and provide a proof of a conjecture of Cerveau [7] . [ABSTRACT FROM AUTHOR]