Geodesic
Vector fields from locally invertible polynomial maps in $\mathbb C^n$
Alvaro Bustinduy 1   ; Luis Giraldo 2   ; Jesús Muciño-Raymundo 3
1 Departamento de Ingeniería Industrial Escuela Politécnica Superior Universidad Antonio de Nebrija C/ Pirineos 55 28040 Madrid, Spain
2 Instituto de Matemática Interdisciplinar (IMI) Departamento de Geometría y Topología Facultad de Ciencias Matemáticas Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid, Spain
3 Centro de Ciencias Matemáticas UNAM, Campus Morelia A.P. 61-3 (Xangari) 58089 Morelia, Michoacán, México
Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 205-220 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/cm140-2-4
Keywords: ldots mathbb mathbb locally invertible polynomial map consider canonical pull back vector fields under map denoted partial partial ldots partial partial main result following n vector fields partial partial have complete holomorphic flows along typical fibers submersion ldots ldots inverse map exists several equivalent versions main hypothesis given

Alvaro Bustinduy  1   ; Luis Giraldo  2   ; Jesús Muciño-Raymundo  3

1 Departamento de Ingeniería Industrial Escuela Politécnica Superior Universidad Antonio de Nebrija C/ Pirineos 55 28040 Madrid, Spain
2 Instituto de Matemática Interdisciplinar (IMI) Departamento de Geometría y Topología Facultad de Ciencias Matemáticas Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid, Spain
3 Centro de Ciencias Matemáticas UNAM, Campus Morelia A.P. 61-3 (Xangari) 58089 Morelia, Michoacán, México 
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     title = {Vector fields from locally invertible
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     journal = {Colloquium Mathematicum},
     pages = {205--220},
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     doi = {10.4064/cm140-2-4},
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Alvaro Bustinduy; Luis Giraldo; Jesús Muciño-Raymundo. Vector fields from locally invertible
 polynomial maps in $\mathbb C^n$. Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 205-220. doi: 10.4064/cm140-2-4

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