Vector fields with simply connected trajectories transverse to a polynomial
This paper considers certain holomorphic vector fields on \(\mathbb{C}^2\) that have no zeroes. By prior results, up to (polynomial or holomorphic, depending on the circumstances and the desired simplicity) change of coordinates, such a vector field \(X\) has a simple polynomial form whenever all of its trajectories are isomorphic to \(\mathbb{C}\) and have analytic closures. This paper produces similarly simple forms under different assumptions on \(X\) -- most importantly, that all of the trajectories of \(X\) are simply connected. As noted in the paper, the assumptions on \(X\) are somewhat limiting; however, they are actually a clever way of accounting for every \(X\) whose associated derivation \[ D_X : f \mapsto X(f) \] (on \(\mathbb{C}[x,y]\)) is surjective. While the flavor of the proofs in the paper does not depart substantially from that of previous work in the field, the paper makes many improvements to the details of the understanding of the vector fields considered in order to achieve the main result.
scientific article; zbMATH DE number 6499696