Effective computation of centralizers of ODOs
Antonio Jiménez-Pastor, Sonia L. Rueda
TL;DR
This work addresses the problem of computing the centralizer $Z(L)$ of a differential operator $L$ in the differential-operator ring, aiming to output a basis of $Z(L)$ as a $C[L]$-module and to generate families of algebro-geometric ODOs via level varieties. The authors combine Goodearl’s finite-basis structure with solving linearized stationary Gelfand-Dickey hierarchies after specializing $L$’s coefficients, and extend the framework to differential fields with parametric coefficients. They introduce a filtration-based, optimized approach that yields a filtered basis $\{1,G_1^*,\ldots,G_{d-1}^*\}$ and a maximal submodule $\mathcal{M}_{\rho}$ when gcd$(n,M)\neq 1$, along with a hyperelliptic-extendable, linearly computable setting to handle coefficient fields; they also demonstrate extensive computations for hyperbolic, elliptic, and rational coefficient families and provide a SageMath implementation (dalgebra) to reproduce the results. The paper thus builds a concrete algorithmic bridge between the algebraic-geometry of spectral curves and computable centralizers of ODOs, enabling the construction and exploration of nontrivial centralizers and algebro-geometric families in practical settings. Overall, the approach advances computational differential algebra by delivering (i) a Goodearl-based basis computation, (ii) GD-hierarchy–driven linearization, (iii) level-variety–driven family generation, and (iv) reproducible software support for hyperbolic, elliptic, and rational coefficient cases.
Abstract
This work is devoted to computing the centralizer $Z (L)$ of an ordinary differential operator (ODO) in the ring of differential operators. Non-trivial centralizers are known to be coordinate rings of spectral curves and contain the ring of polynomials $C [L]$, with coefficients in the field of constants $C$ of $L$. We give an algorithm to compute a basis of $Z (L)$ as a $C [L]$-module. Our approach combines results by K. Goodearl in 1985 with solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchy, which after substituting the coefficients of $L$ become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of $L$ belong to a differential algebraic extension $K$ of $C$. In addition, by considering parametric coefficients we develop an algorithm to generate families of ODOs with non trivial centralizer, in particular algebro-geometric, whose coefficients are solutions in $K$ of systems of the stationary GD hierarchy.