Two Results on Homogeneous Hessian Nilpotent Polynomials
| dc.creator | Essen, Arno van den | |
| dc.creator | Zhao, Wenhua | |
| dc.date | 2007-04-13 | |
| dc.date.accessioned | 2026-07-07T12:35:52Z | |
| dc.date.available | 2026-07-07T12:35:52Z | |
| dc.description | Let $z=(z_1, ..., z_n)$ and $Δ=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be {\it Hessian Nilpotent}(HN) if its Hessian matrix $\Hes P(z)=(\frac {\partial^2 P}{\partial z_i\partial z_j})$ is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $Δ^m P^{m+1}(z)=0$ for any $m>>0$.} In this paper, we first show that, the VC holds for any homogeneous HN polynomial $P(z)$ provided that the projective subvarieties ${\mathcal Z}_P$ and ${\mathcal Z}_{σ_2}$ of $\mathbb C P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $σ_2(z):=\sum_{i=1}^n z_i^2$, respectively, intersect only at regular points of ${\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z-\nabla P$ with $P(z)$ HN if $F$ has no non-zero fixed point $w\in \mathbb C^n$ with $\sum_{i=1}^n w_i^2=0$. Secondly, we show that the VC holds for a HN formal power series $P(z)$ if and only if, for any polynomial $f(z)$, $Δ^m (f(z)P(z)^m)=0$ when $m>>0$. | |
| dc.description | Latex, 7 pages | |
| dc.identifier | https://arxiv.org/abs/0704.1690 | |
| dc.identifier | http://arxiv.org/abs/0704.1690 | |
| dc.identifier | J. Pure Appl. Algebra, 212 (2008), no. 10, 2190--2193. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/217905 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Complex Variables | |
| dc.subject | 14R15, 31B05 | |
| dc.title | Two Results on Homogeneous Hessian Nilpotent Polynomials | |
| dc.type | text |