Hessian nilpotent polynomials and the Jacobian conjecture
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- by Wenhua Zhao PDF
- Trans. Amer. Math. Soc. 359 (2007), 249-274 Request permission
Abstract:
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.References
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Additional Information
- Wenhua Zhao
- Affiliation: Department of Mathematics, Illinois State University, Normal, Illinois 61790-4520
- Email: wzhao@ilstu.edu
- Received by editor(s): October 15, 2004
- Received by editor(s) in revised form: October 26, 2004
- Published electronically: July 20, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 249-274
- MSC (2000): Primary 33C55, 39B32, 14R15, 31B05
- DOI: https://doi.org/10.1090/S0002-9947-06-03898-0
- MathSciNet review: 2247890