A new sufficient condition in order that the Real Jacobian Conjecture in ℝ² holds

Let ( f , g ) : R 2 → R 2 (f, g) : {\mathbb {R}}^2\to {\mathbb {R}}^2 be a polynomial map satisfying that its Jacobian is different from zero at all points p ∈ R 2 p\in {\mathbb {R}}^2 and ( f , g ) ( 0 , 0 ) = ( 0 , 0 ) (f, g) (0, 0) = (0, 0) . Then the Real Jacobian Conjecture states that the polynomial map ( f , g ) (f, g) is injective. Pinchuk [Math. Z. 217 (1994), pp. 1–4] proved in 1994 that this conjecture is false. Now several authors are working providing some additional condition to the conjecture in order that this holds. In this paper we prove that if the homogeneous part of higher degree of the polynomial f 2 + g 2 f^2 + g^2 does not have real linear factors, then the map ( f , g ) (f, g) is injective. This result is proved using the qualitative theory of the polynomial differential systems in the plane R 2 {\mathbb {R}}^2 .