Hessian nilpotent polynomials and the Jacobian conjecture

Let $z=(z_1,\cdots,z_n)$ and let $\Delta=\sum_{i=1}^n \frac{\partial^2}{\partial z^2_i}$ be 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 what we call the 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 $m> \frac{3}{2}(3^{n-2}-1)$. 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 $\mathrm{Hes}\,P(z)=(\frac{\partial^2 P}{\partial z_i\partial z_j})$ is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen 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.