by  A.V.Domrina  and  S.Yu. Orevkov

Abstract:   Let  f : C² -> C²  be a polynomial mapping whose jacobian is equal to one. The well known Jacobian Conjecture states that  f  must be a one-to-one mapping.  Let  N  be the topological degree of f  (i.e. the number of preimages of a generic point). The Jacobian Conjecture is equivalent to the fact that  N =1. It is proved in  [O]  that  N  cannot be equal to  2  or  3.

In this paper we give the first part of the proof of the following result:

  Theorem 1.  N  cannot be equal to  4.

Namely, we prove Theorem 1 in the case when  f  has only one dicritical component, i.e. a component of the complement of  C²  in  X  which is not mapped to a single point (here  X -> C²  is an extension of  f  up to a proper mapping of complex manifolds). It follows from  [O]  that  f  may have one or two dicritical components when  N =4.

The proof is based on the comparing of the graphs of compactifications of  X  and C² where the compactifications are chosen so that  f  extends to a regular mapping and the image of the dicritical component is transversal to the curve at infinity.

In the case when  f  has two dicritical components, Theorem 1 is proved in  [D]  using the same techniques.

[D]  A.V.Domrina,  Four-sheeted polynomial mappings of  C². The general case.  Izvestiya. Math. 64(2000), 1-33
[O]  S.Yu.Orevkov,   On three-sheeted polynomial mappings of C². Math. USSR-Izvestiya,  29(1987), 587-596