Gr\"obner bases of Burchnall-Chaundy ideals for ordinary differential operators

Abstract

The correspondence between commutative rings of ordinary differential operators (ODOs) and algebraic curves was established by Burchnall and Chaundy, Krichever and Mumford, among many others. To make this correspondence computationally effective, in this paper we aim to compute the defining ideals of spectral curves, Burchnall-Chaundy (BC) ideals. We provide an algorithm to compute a Gr\"obner basis of a BC ideal. The point of departure is the computation of the finite set of generators of a maximal commutative ring of ODOs, which was implemented by the authors in the package dalgebra of SageMath. The algorithm to compute BC ideals has been also implemented in dalgebra. The differential Galois theory of the corresponding spectral problems, linear differential equations with parameters, would benefit from the computation on this prime ideal, generated by constant coefficient polynomials. In particular, we prove the primality of the differential ideal generated by a BC ideal, after extending the coefficient field. This is a fundamental result to develop Picard-Vessiot theory for spectral problems.