Gröbner bases of Burchnall-Chaundy ideals for ordinary differential operators
Antonio Jiménez-Pastor, Sonia L. Rueda
TL;DR
The paper addresses the computation of spectral curves for commutative rings of ordinary differential operators by deriving Burchnall-Chaundy ideals and computing their Gröbner bases via a Goodearl basis of the centralizer $\mathcal{Z}(L)$. It introduces a structured set of relations $R_{i,j}(\lambda,\overline{\mu})$ that form a Gröbner basis for $\mathrm{BC}(L)$ under a weighted order, yielding the coordinate ring $\mathbf{C}[\Gamma]=\mathbf{C}[\lambda,\overline{\mu}]/\mathrm{BC}(L)$ and enabling effective exploration of the algebraic and differential structure. A key theoretical contribution is proving that the extended BC differential ideal $[\mathrm{BC}(L)]$ is prime, which produces a differential domain $\mathbb{K}[\Gamma]$ and a Picard-Vessiot framework for spectral problems of arbitrary order. The results are complemented by concrete algorithms and an implementation in the SageMath dalgebra package, making the spectral-curve construction practical for applications in integrable systems and differential Galois theory. The work thus bridges algebraic geometry, differential algebra, and computational methods to enable robust analysis of spectral problems associated with 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 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.