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Hessian Nilpotent Polynomials and the Jacobian Conjecture | | Wenhua Zhao
; | | Date: |
27 Sep 2004 | | Subject: | Complex Variables; Algebraic Geometry MSC-class: 33C55, 39B32, 14R15, 31B05 | math.CV math.AG | | Abstract: | Let $z=(z_1, ..., z_n)$ and $Delta=sum_{i=1}^n fr {p^2}{z^2_i}$ 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 $Hes P(z)=(fr {p^2 P}{z_iz_j})$ 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. | | Source: | arXiv, math.CV/0409534 | | Services: | Forum | Review | PDF | Favorites | |
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