关于常数微分算子的一个消失猜想 系数

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摘要翻译：
在[BE1]、[Me]和[Z2]的最新进展中，已将著名的JC（Jacobian猜想）([BCW],[E])归结为关于Laplace算子和HN（Hessian幂零）多项式（Hessian矩阵为幂零的多项式）的VC（消失猜想）。本文首先证明了对于所有二阶齐次微分算子$\lambda$和$\lambda$-幂零多项式($p(z)$满足$\lambda^m p^m=0$对于所有$m\ge1$的多项式），上述消失猜想和JC等价于一个消失猜想。然后，我们将文献中关于JC、HN多项式和Laplace算子的VC的一些结果转化为关于二阶齐次微分算子的$\lambda$-幂零多项式及其相关的VC的某些结果。本文的这一部分也可以作为一个关于HN多项式和在更一般的情况下相关的VC的简短综述来读。最后，我们讨论了在一个或多个变量中，$λ$-幂零多项式与经典正交多项式之间的一个尚待理解的联系。这种联系为二阶齐次满秩常系数微分算子λ-幂零多项式的各向同性提供了概念上的理解。
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英文标题：
《A Vanishing Conjecture on Differential Operators with Constant
  Coefficients》
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作者：
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：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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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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英文摘要：
  In the recent progress [BE1], [Me] and [Z2], the well-known JC (Jacobian conjecture) ([BCW], [E]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd order homogeneous differential operators $\Lambda$ and $\Lambda$-nilpotent polynomials $P$ (the polynomials $P(z)$ satisfying $\Lambda^m P^m=0$ for all $m\ge 1$). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on $\Lambda$-nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators $\Lambda$. This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of $\Lambda$-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous $\Lambda$-nilpotent polynomials for the 2nd order homogeneous full rank differential operators $\Lambda$ with constant coefficients.
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PDF链接：
https://arxiv.org/pdf/0704.1691

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关键词：微分算子 Differential coefficients Polynomials homogeneous 综述 算子 尚待 nilpotent 概念