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

TL;DR

This paper develops an algorithm to compute Gr"obner bases of Burchnall-Chaundy ideals, linking differential operators and algebraic curves, with applications to spectral problems and differential Galois theory.

Contribution

It introduces an algorithm for computing BC ideals' Gr"obner bases and proves the primality of the associated differential ideal, enhancing spectral problem analysis.

Findings

Algorithm successfully computes BC ideals' Gr"obner bases.

Proves primality of the differential ideal generated by BC ideals.

Implementation available in SageMath's dalgebra package.

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.