摘要翻译：
设$z=(z_1,...,z_n)$和$\delta=\sum_{i=1}^n\frac{\部分^2}{\部分Z^2_i}$为拉普拉斯运算符。如果形式幂级数$P(z)$的Hessian矩阵$Hes P(z)=(\frac{\partial^2P}{\partial Z_i\partial z_j}）$是幂零的，则称其为{\it Hessian幂零}(HN)。在[BE1]、[M]和[Z]中，雅可比猜想被归结为以下所谓的HN多项式的{It消失猜想}(VC)：{It对于任意齐次HN多项式$p(Z)$$($d=4$$)$,对于任意$M>>0$,我们有$\delta^M p^{M+1}(Z)=0$本文首先证明了只要由$p(z)$和$\sigma2(z):=\sum_{i=1}^n z_i^2$生成的主理想决定的$\mathbb C p^{n-1}$的射影子簇${\mathcal z}_p$和${\mathcal z}_{\sigma2}$仅相交于${mathcal z}_p$的正则点，VC对任何齐次HN多项式$p(z)$成立。因此，对于对称多项式映射$F=Z-\nabla P$与$P(z)$HN,如果$F$不存在非零不动点$W\in\mathBB C^n$与$\sum_{i=1}^nw_i^2=0$,则Jacobian猜想成立。其次，我们证明了对于任意多项式$f(z)$,当$m>0$时，VC对于HN形式幂级数$P(z)$成立当且仅当$\delta^m(f(z)P(z)^m)=0$。
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英文标题：
《Two Results on Homogeneous Hessian Nilpotent Polynomials》
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作者：
Arno van den Essen and Wenhua Zhao
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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英文摘要：
  Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ 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 $\Delta^m P^{m+1}(z)=0$ 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 ${\mathcal Z}_{\sigma_2}$ of $\mathbb C P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $\sigma_2(z):=\sum_{i=1}^n z_i^2$, 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 $f(z)$, $\Delta^m (f(z)P(z)^m)=0$ when $m>>0$.
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PDF链接：
https://arxiv.org/pdf/0704.1690