The Real Jacobian Conjecture for polynomials of degree 3 (Q2724125)

The paper solves the famous real Jacobian conjecture in the following particular case: NEWLINENEWLINENEWLINEIf \((f,g):\mathbb R^2\to\mathbb R^2\) is a polynomial mapping such that \(\deg f\leq 3\), \(\deg g\leq 3\) and the Jacobian \(\text{Jac}(f,g)\) of \((f,g)\) is strictly positive in \(\mathbb R^2\), i.e. \(\text{Jac}(f,g)(x)>0\) for every \(x\in \mathbb R^2\), then \((f,g)\) is a global diffeomorphism. NEWLINENEWLINENEWLINENotice that the above is not true if \(\deg f\), \(\deg g \) are sufficiently large (10 and 35, respectively, in the Pinchuk example) and that it is not assumed that \(\text{Jac}(f,g)\) is constant.