Abstract:

Let $F=(f,g): mathbb{R}^2tomathbb{R}^2$ be a polynomial map such that $det DF(x)$ is different from zero for all $x$ in $mathbb{R}^2$. 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. In this talk we shall prove that for the injectivity of $F$ it is sufficient to assume that there exists at least one point $a$ in $mathbb{R}^2$ such that the cardinal of $F^{-1}(a)$ is one, and that $f^*$ and $g^*$ do not have real linear factors in common.