Vector fields with simply connected trajectories transverse to a polynomial
type: Item
instance of: scholarly article
title: Vector fields with simply connected trajectories transverse to a polynomial
zbMATH Open document ID: 1337.32034
author: Alvaro Bustinduy
author: Luis Giraldo
published in: Advances in Mathematics
publication date: 2015-10-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32M25
Mathematics Subject Classification ID: 32S65
zbMATH DE Number: 6499696
zbMATH Keywords: holomorphic vector fields
zbMATH Keywords: foliation transverse to a fibration
zbMATH Keywords: trajectories
zbMATH Keywords: surjective derivations
zbMATH Keywords: derivations with slices
MaRDI profile type: Publication
full work available at URL: j.aim.2015.08.003
OpenAlex ID: W1439441293
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DOI: 10.1016/J.AIM.2015.08.003
description: scientific article; zbMATH DE number 6499696