  Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \fr {\p^2}{\p z^2_i}$ the
Laplace operator. The main goal of the paper is to show that the well-known
Jacobian conjecture without any additional conditions is equivalent to the
following what we call {\it vanishing conjecture}: for any homogeneous
polynomial $P(z)$ of degree $d=4$, if $\Delta^m P^m(z)=0$ for all $m \geq 1$,
then $\Delta^m P^{m+1}(z)=0$ when $m>>0$, or equivalently, $\Delta^m
P^{m+1}(z)=0$ when $m> \fr 32 (3^{n-2}-1)$. It is also shown in this paper that
the condition $\Delta^m P^m(z)=0$ ($m \geq 1$) above is equivalent to the
condition that $P(z)$ is Hessian nilpotent, i.e. the Hessian matrix $\Hes
P(z)=(\fr {\p^2 P}{\p z_i\p z_j})$ is nilpotent. The goal is achieved by using
the recent breakthrough work of M. de Bondt, A. van den Essen \cite{BE1} and
various results obtained in this paper on Hessian nilpotent polynomials. Some
further results on Hessian nilpotent polynomials and the vanishing conjecture
above are also derived.
