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

${\rm Jacobian Conjecture}_{2n}$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。

Jacobian猜想与Dixobian猜想是稳定等价的

The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture

关于$n$维的雅可比猜想断言，在特征为零的域上，$n$维仿射空间的任何多项式自同态，当雅可比等于1时，都是可逆的。秩为$n$的Dixandian猜想断言$n$阶Weyl代数（$n$变量中的多项式微分算子代数）的任何自同态都是可逆的。证明了维度$2n$上的Jacobian猜想蕴含秩$n$上的Dixmier猜想。再加上一个著名的相反方向的含义，它表明稳定的雅可比矩阵和Dixobian矩阵是等价的。证明的主要工具是有限特征化约。在论文完成后，我们了解到主要结果已经由Y.Tsuchimoto在Osaka Journal of Mathematics第42卷第2期（2005年6月）上发表。他的证明是不同的。

Poisson代数的相对Dixmier性质

Relative Dixmier property for Poisson algebras

本文提出Poisson代数的相对Dixmier性质，用于证明若干类Poisson代数具有Dixmier性质，并应用于取消问题和Hopf余作用的非存在性等研究。

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…

Poisson algebras via model theory and differential-algebraic geometry

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are…

Model theory and the DME: a survey

Recent work using the model theory of differentially closed fields to answer questions having to do with the Dixmier-Moeglin equivalence for (noncommutatve) finitely generated noetherian algebras, and for (commutative) finitely generated Poisson algebras, is here surveyed, with an emphasis on the model-theoretic and differential-algebraic-geometric antecedents.