Jacobian猜想与Dixobian猜想是稳定等价的

${\rm Jacobian Conjecture}_{2n}$意味着${\rm Dixmier Problem}_n$

The ${\rm Jacobian Conjecture}_{2n}$ implies the ${\rm Dixmier Problem}_n$

利用作者给出的多项式系数Weyl代数自同构的反演公式及其次数的界，对A. Belov-Kanel和M. Kontsevich证明了Jacobian猜想蕴含着Dixobian问题。没有原创性。

关于Jacobian猜想、Dixandian猜想和Poisson猜想在任意特征下的等价性

On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic

借助于Azumaya代数的环同态的最新结果和随后关于规范Poisson代数和Dirac量子代数的自同态的结果，以及关于这些代数在特征零下的正特征的重构，我们证明了这三个代数在任何特征下的等价性，用这种方法也给出了Y.土本

Holonomic modules and 1-generation in the Jacobian Conjecture

A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result h…

关于$A_1$的星型Dixandian猜想

The starred Dixmier conjecture for $A_1$

设$A_1(K)=K \langle x,y | yx-xy= 1 \rangle$是特征零域$K$上的第一Weyl代数，$α$是由$α(x)= y$和$α(y)= x$给出的$A_1(K)$上的交换对合. 1968年，Dixlane的Dixlane猜想提出了一个问题：Weyl代数$A_1(K)$的每一个代数自同态都是自同构吗？证明了$A_1(K)$的每个$α$ -自同态都是自同构.这里$A_1(K)$的$α$ -自同态是保持对合$α$的自同态.我们还证明了二维Jacobian猜想的一个类似结果，称为$α-JC_2$。

Weyl代数中元素的正规形式与Dixmier猜想

Normal forms of elements in the Weyl algebra and Dixmier Conjecture

本文研究Weyl代数中元素的Joseph正规形式，证明其对应唯一整数对(k,n)，并推广Dixmier猜想在k或n为素数时的成立条件，同时获得Jacobian猜想的类似结果。

A Dixmier theorem for Poisson enveloping algebras

We consider a skew-symmetric $n$-ary bracket on the polynomial algebra $K[x_1,\ldots,x_n,x_{n+1}]$ ($n\geq 2$) over a field $K$ of characteristic zero defined by $\{a_1,\ldots,a_n\}=J(a_1,\ldots,a_n,C)$, where $C$ is a fixed element of $K[x_1,\ldots,x_n,x_{n+1}]$ and $J$ is the Jacobian. If $n=2$ then this bracket is a Poisson bracket and if $n\geq 3$ then it is an $n$-Lie-Poisson bracket on $K[x_1,\ldots,x_n,x_{n+1}]$. We describe the center of the corresponding $n$-Lie-Poisson algebra and show that the quotient algebra $K[x_1,\ldots,x_n,x_{n+1}]/(C-λ)$, where $(C-λ)$ is the ideal generated b…