# On solvable elements in the Weyl algebra

## TL;DR

This paper explores the properties of solvable elements in the Weyl algebra and relates their classification to the open Dixmier problem, providing conditions for unsolvability and analyzing their characteristics.

## Contribution

It introduces new sufficient conditions for elements to be unsolvable in the Weyl algebra and studies properties of solvable elements, advancing understanding of the Dixmier problem.

## Key findings

- Sufficient conditions for unsolvability are established.
- Properties of solvable elements are characterized.
- The work relates solvability to the Dixmier open question.

## Abstract

We show that to determine all solvable elements in the Weyl algebra is closely related to the Dixmier's open question. Sufficient conditions for an elements being unsolvable are given, and properties of solvable elements are obtained.