Abstract This work is devoted to the symbolic computation of centralizers of ordinary differential operators (ODOs), in the ring of differential operators. Starting with an operator L L of order n n and the order m m of a non-trivial operator in its centralizer, which is not a multiple of n n, a finite set of generators of a subalgebra of the centralizer is obtained, maximal of a certain rank R R, the greatest common divisor of all orders of its elements. The true rank r r of the centralizer is unknown to start unless (n, m) =1 (n, m) = 1 since 1 r R (n, m) 1 ≤ r ≤ R ≤ (n, m), and remains unknown unless our algorithm returns R=1 R = 1. Ours is a direct approach based on solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchies, which after substituting the coefficients of L L become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of L L belong to a computable differential field. In addition, by considering parametric coefficients, we develop an algorithm to generate families of ODOs with non-trivial centralizer, whose coefficients belong to a previously chosen differential field. Our algorithms are implemented in SageMath.
Jiménez-Pastor et al. (Mon,) studied this question.