Sufficient conditions in order that the real Jacobian conjecture in $\mathbb{R}^2$ holds [ Back ]

Date:
14.10.13   
Times:
15:30 to 16:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Jaume Llibre
University:
Universitat Autònoma de Barcelona

Abstract:

Let $F=(f,g): \mathbb{R}^2\to\mathbb{R}^2$ 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.