A sufficient condition in order that the real Jacobian conjecture in R^2 holds

Let F=(f,g):R2→R2 be a polynomial map such that det⁡DF(x,y) is different from zero for all (x,y)∈R2 and F(0,0)=(0,0). We prove that for the injectivity of F it is sufficient to assume that the higher homogeneous terms of the polynomials ffx+ggx and ffy+ggy do not have real linear factors in common. The proofs are based on qualitative theory of dynamical systems.The first author is partially supported by a BPE-FAPESP grant number 2014/26149-3. The second author is partially supported by a MINECO/FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The third author is partially supported by a MINECO grant number MTM2013- 40998-P, and AGAUR grant number 2014SGR 568 and two FP7-PEOPLE- 2012-IRSES grants numbers 316338 and 318999. The first and the third author are also partially supported by a CAPES CSF–PVE grant 88881. 030454/ 2013-01 from the program CSF-PVE