Vector fields from locally invertible polynomial maps in $\mathbb C^n$

Volume 140 / 2015

Alvaro Bustinduy, Luis Giraldo, Jesús Muciño-Raymundo Colloquium Mathematicum 140 (2015), 205-220 MSC: Primary 14R15; Secondary 37F75, 32M17, 32M25. DOI: 10.4064/cm140-2-4


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.


  • Alvaro BustinduyDepartamento de Ingeniería Industrial
    Escuela Politécnica Superior
    Universidad Antonio de Nebrija
    C/ Pirineos 55
    28040 Madrid, Spain
  • Luis GiraldoInstituto de Matemática Interdisciplinar (IMI)
    Departamento de Geometría y Topología
    Facultad de Ciencias Matemáticas
    Universidad Complutense de Madrid
    Plaza de Ciencias 3
    28040 Madrid, Spain
  • Jesús Muciño-RaymundoCentro de Ciencias Matemáticas
    UNAM, Campus Morelia
    A.P. 61-3 (Xangari) 58089
    Morelia, Michoacán, México

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