On Some Questions around Berest’s Conjecture

Let \(K\) be a field of characteristic zero, and let \(A_1=K[x][\partial ]\) be the first Weyl algebra. In the present paper, we prove the following two results.

\(\bullet\) Assume that there exists a nonzero polynomial \(f(X,Y)\in K[X,Y]\) such that (i) \(f\) has a nontrivial solution \((P,Q)\in A_{1}^{2}\) with \([P,Q]=0\) ; (ii) the set of solutions of \(f\) in \(A_{1}^{2}\) splits into finitely many \(\operatorname{Aut}(A_1)\) -orbits under the natural actuon of the group \(\operatorname{Aut}(A_1)\) . Then the Dixmier conjecture holds; i.e., every \(\varphi\in \operatorname{End}(A_{1})\setminus\{0\}\) is an automorphism.

\(\bullet\) Assume that \(\varphi\in \operatorname{End}(A_{1})\) is an endomorphism of monomial type. (In particular, it is not an automorphism; see Theorem 4.1.) Then \(\varphi\) has no nontrivial fixed points; i.e. there exists no \(P\in A_1\setminus K\) such that \(\varphi (P)=P\) .