Complete polynomial vector fields on the complex plane.
type: Item
instance of: scholarly article
title: Complete polynomial vector fields on the complex plane.
zbMATH Open document ID: 1047.32015
DOI: 10.1016/S0040-9383(03)00050-8
author: Marco Brunella
published in: Topology
publication date: 2004-03-14T00:00:00Z
review text: 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.
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 32M25
Mathematics Subject Classification ID: 34M45
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 2056783
zbMATH Keywords: polynomial vector field
zbMATH Keywords: two complex variables
zbMATH Keywords: complex plane
zbMATH Keywords: complete vector field
zbMATH Keywords: complete flow
zbMATH Keywords: polynomial fibration
zbMATH Keywords: holomorphic foliation
zbMATH Keywords: projective surface
zbMATH Keywords: polynomial automorphism
zbMATH Keywords: minimal model
zbMATH Keywords: Mori theory
MaRDI profile type: Publication
full work available at URL: s0040-9383(03)00050-8
OpenAlex ID: W2060822900
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description: scientific article; zbMATH DE number 2056783