Jacobian mates for non-singular polynomial maps in C^n with one-dimensional fibers

Journal of Singularities
volume 9 (2014), 27-42

Received 27 May 2013. Received in revised form 15 November 2013.

DOI: 10.5427/jsing.2014.9b

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Abstract:

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).

Keywords:

Dominating polynomial maps, Jacobian conjecture, non-singular polynomial vector fields, Abelian integrals

Mathematical Subject Classification:

14R15 (primary), 37F75 (secondary)

Author(s) information: