On some questions around Berest's conjecture

TL;DR

This paper explores conditions under which the Dixmier conjecture holds for the Weyl algebra, linking solutions of polynomial equations and automorphism properties, and investigates fixed points of certain endomorphisms.

Contribution

It establishes that specific conditions on solutions of polynomial equations imply the Dixmier conjecture, and analyzes fixed points of monomial-type endomorphisms.

Findings

If a polynomial with a finite orbit of solutions exists, the Dixmier conjecture holds.

Monomial-type endomorphisms have no non-trivial fixed points.

Conditions linking polynomial solutions to automorphism conjectures are identified.

Abstract

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 $\forall \varphi\in End(A_{1})-\{0\}$, $\varphi$ is an automorphism. Assume $\varphi$ 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. $\varphi (P)=P$.