**T. Asanuma**

Toyama University, Japan

March 17, 2011

**On the Jacobian conjecture**:
Let $\phi=(f,g):{\bf C}^2\rightarrow{\bf C}^2$ be a polynomial map of the plane over the field {\bf C} of complex numbers with its Jacobian nonzero constant. The Jacobian conjecture asserts that $\phi$ must be an automporphism.
We say that a point ~$P_{\infty} \in {\bf P}^2 \backslash {\bf C}^2$
of the complex projective plane ${\bf P}^2$ is ``quasifinite'' (w.r.t. $\phi$) if there exists a sequence $\{P_i\}$ in ${\bf C}^2 \subset {\bf P}^2$ converging to $P_\infty$ such that the image $\{\phi(P_i)\}$ converges to a point in ${\bf C}^2$. In this talk I will show that the conjecture holds if and only if there is no quasifinite point.