Gröbner bases of Burchnall-Chaundy ideals for ordinary differential operators

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 Gröbner 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.

💡 Research Summary

The paper addresses the long‑standing problem of effectively computing the defining ideals of spectral curves associated with maximal commutative subrings of ordinary differential operators (ODOs). Building on the classical Burchnall‑Chaundy correspondence, which links commuting ODOs to algebraic curves, the authors develop a concrete algorithm that produces a Gröbner basis for the Burchnall‑Chaundy (BC) ideal of a given operator (L).

The starting point is the previously implemented routine in the SageMath package dalgebra that, for a differential operator (L) with a non‑trivial centralizer, computes a finite (C