Open AccessOn some questions around Berest's conjectureOpen Access
Author(s)
Junhu Guo,
Alexander Zheglov
Publication year2024
Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be thefirst 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 anon-trivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$, and the number oforbits 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 $\forall \varphi\inEnd(A_{1})-\{0\}$, $\varphi$ is an automorphism. Assume $\varphi$ is an endomorphism of monomial type (in particular, it isnot 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. $\varphi (P)=P$.
Language(s)English
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