Geodesic
Jacobian mates for non-singular polynomial maps in C^n with one-dimensional fibers
Journal of Singularities, Tome 9 (2014), pp. 27-42

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Let (F_2, ... , F_n): C^n -> C^{n-1}$ be a non-singular polynomial map. We introduce a non-singular polynomial vector field X tangent to the foliation F having as leaves the fibers of the map (F_2, ... , F_n). Suppose that the fibers of the map are irreducible in codimension at least 2, that the one forms of time associated to the vector field X are exact along the leaves, and that there is a finite set at the hyperplane at infinity containing all the points necessary to compactify the affine curves appearing as fibers of the map. Then, there is a polynomial F_1 (a Jacobian mate) such that the completed map (F_1, F_2, ... , F_n) is a local biholomorphism. Our proof extends the integration method beyond the known case of planar curves (introduced by Ilyashenko).
DOI : 10.5427/jsing.2014.9b
Classification : 14R15(primary), 37F75(secondary) 
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     author = {Alvaro Bustinduy and Luis Giraldo, and Jes\'us Muci\~no-Raymundo},
     title = {Jacobian mates for non-singular polynomial maps in {C^n} with one-dimensional fibers},
     journal = {Journal of Singularities},
     pages = {27--42},
     publisher = {mathdoc},
     volume = {9},
     year = {2014},
     doi = {10.5427/jsing.2014.9b},
     url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2014.9b/}
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Alvaro Bustinduy; Luis Giraldo,; Jesús Muciño-Raymundo. Jacobian mates for non-singular polynomial maps in C^n with one-dimensional fibers. Journal of Singularities, Tome 9 (2014), pp. 27-42. doi: 10.5427/jsing.2014.9b

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