The phase portrait of the Hamiltonian system associated to a Pinchuk map.

In this paper we describe the global phase portrait of the Hamiltonian system associated to a Pinchuk map in the Poincaré disc. In particular, we prove that this phase portrait has 15 separatrices, five of them singular points, and 7 canonical regions, six of them of type strip and one annular.

A new qualitative proof of a result on the real Jacobian conjecture.

Let F= (f, g) : R2 → R2 be a polynomial map such that det DF(x) is different from zero for all x ∈ R2. We assume that the degrees of f and g are equal. We denote by f and G the homogeneous part of higher degree of f and g, respectively.