Vector fields from locally invertible polynomial maps in $\mathbb C^n$
Volume 140 / 2015
Colloquium Mathematicum 140 (2015), 205-220
MSC: Primary 14R15; Secondary 37F75, 32M17, 32M25.
DOI: 10.4064/cm140-2-4
Abstract
Let $(F_1, \ldots , F_n): \mathbb {C}^n \to \mathbb {C}^{n}$ be a locally invertible polynomial map. We consider the canonical pull-back vector fields under this map, denoted by $\partial / \partial F_1, \ldots , \partial / \partial F_n $. Our main result is the following: if $n-1$ of the vector fields $\partial / \partial F_j$ have complete holomorphic flows along the typical fibers of the submersion $(F_1, \ldots , F_{j -1}, F_{j+1} , \ldots , F_n)$, then the inverse map exists. Several equivalent versions of this main hypothesis are given.
