# Commuting Ordinary Differential Operators and the Dixmier Test

## TL;DR

This paper investigates the structure of commutative subalgebras in the Weyl algebra, introduces the Dixmier test for centralizers, and constructs Burchnall-Chaundy pairs for order 4 operators, advancing computational methods in differential algebra.

## Contribution

It develops a computational approach to identify centralizers and construct BC pairs in the Weyl algebra, addressing a longstanding open problem.

## Key findings

- Established the Dixmier test for centralizer conditions.
- Constructed BC pairs for order 4 operators.
- Effectively computed spectral sheaf fibers using differential subresultants.

## Abstract

The Burchnall-Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator $L$ in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator $M$ to be in the centralizer of $L$. Whenever the centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$ pairs, by means of differential subresultants, we effectively compute the fiber of the rank $r$ spectral sheaf over their spectral curve.