Hessian Nilpotent Polynomials and the Jacobian Conjecture
| dc.creator | Zhao, Wenhua | |
| dc.date | 2004-09-27 | |
| dc.date | 2004-11-02 | |
| dc.date.accessioned | 2026-07-07T12:36:37Z | |
| dc.date.available | 2026-07-07T12:36:37Z | |
| dc.description | Let $z=(z_1, ..., z_n)$ and $Δ=\sum_{i=1}^n \fr {\p^2}{\p z^2_i}$ 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 $Δ^m P^m(z)=0$ for all $m \geq 1$, then $Δ^m P^{m+1}(z)=0$ when $m>>0$, or equivalently, $Δ^m P^{m+1}(z)=0$ when $m> \fr 32 (3^{n-2}-1)$. It is also shown in this paper that the condition $Δ^m P^m(z)=0$ ($m \geq 1$) 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. | |
| dc.description | Latex, 34 pages | |
| dc.identifier | https://arxiv.org/abs/math/0409534 | |
| dc.identifier | http://arxiv.org/abs/math/0409534 | |
| dc.identifier | Trans. Amer. Math. Soc. 359 (2007), 249-274. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/218154 | |
| dc.subject | Complex Variables | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 33C55, 39B32, 14R15, 31B05 | |
| dc.title | Hessian Nilpotent Polynomials and the Jacobian Conjecture | |
| dc.type | text |