论文标题
关于贝尔最猜想的一些问题
On some questions around Berest's conjecture
论文作者
论文摘要
令$ k $为特征零的字段，令$ a_1 = k [x] [\ partial] $为第一个Weyl代数。在本文中，我们证明了以下两个结果。 假设存在k [x，y] $中的非零多项式$ f（x，y）\，它在a_ {1}^{2} $ in a_ {1}^{2} $中具有$ [p，q] = 0 $的非平凡解决方案$（p，q）\有限。然后，dixmier猜想保持，即在end（a_ {1}） - \ {0 \} $中，$ \forallφ\，$φ$是一种自动晶体。 假设$φ$是单型的内态性（尤其是它不是自动形态，请参见定理4.1）。然后，它没有非平凡的固定点，即a_1 $，$ p \ notin k $，s.t.中没有$ p \。 $φ（p）= p $。
Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_{1}^{2}$ is finite. Then the Dixmier conjecture holds, i.e $\forall φ\in End(A_{1})-\{0\}$, $φ$ is an automorphism. Assume $φ$ is an endomorphism of monomial type (in particular, it is not an automorphism, see theorem 4.1). Then it has no non-trivial fixed point, i.e. there are no $P\in A_1$, $P\notin K$, s.t. $φ(P)=P$.
