Commuting Ordinary Differential Operators and the Dixmier Test
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 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 to be in the centralizer of . Whenever the centralizer equals the algebra generated by and , we call , a Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for operators of order in the first Weyl algebra. Moreover, for true rank pairs, by means of differential subresultants, we effectively compute the fiber of the rank spectral sheaf over their spectral curve.
Cite
@article{arxiv.1902.01361,
title = {Commuting Ordinary Differential Operators and the Dixmier Test},
author = {Emma Previato and Sonia L. Rueda and Maria-Angeles Zurro},
journal= {arXiv preprint arXiv:1902.01361},
year = {2020}
}