Number of homogeneous components of counterexamples to the Dixmier conjecture

Abstract

Assume that $P$ and $Q$ are elements of $A_1$ satisfying $[P,Q]=1$. The Dixmier Conjecture for $A_1$ says that they always generate $A_1$. We show that if $P$ is a sum of not more than~$4$ homogeneous elements of $A_1$ then $P$ and $Q$ generate $A_1$, which generalizes the main result in arXiv:2210.00257.