Complete polynomial vector fields on the complex plane.
Let \(v=P(z,w)\partial/\partial z + Q(z,w)\partial/\partial w\) be a complete polynomial vector field on the complex plane \(\mathbb C^ 2\). The author proves that then \(v\) is, up to a polynomial change of coordinates, of one of the three types specified in the main theorem of his paper. Each of these three types corresponds to the existence of a polynomial fibration on \(\mathbb C^ 2\), given by the levels of \(z\) if \(v\) is of the first type, by the levels of \(z^ m w^ n\) (where \(m,n\in \mathbb N^ +\), \(\langle m,n\rangle=1)\) if \(v\) is of the second type or by the levels of \(z^ m(z^ lw+p(z))^ n\) (where \(m,n,l\in \mathbb N^ +\), \(\langle m,n\rangle=1\), \(p\in \mathbb C[z]\), \(\text{deg}(p)\leq l-1\), \(p(0)\neq 0\)) if \(v\) is of the third type. The proof is mainly based on \textit{M. Suzuki's} work [Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 517--546 (1977; Zbl 0403.32020)] on the classification of complete vector fields admitting a rational first integral, and on M. McQuillan's classification of foliations on projective surfaces with a transcendental entire leaf.
scientific article; zbMATH DE number 2056783
scientific article