Computing almost commuting bases of ODOs and Gelfand-Dikey hierarchies

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dikey (GD) hierarchies. In this paper, we review the formal construction of the vector space of almost commuting operators with a given ordinary differential operator (ODO), with the ultimate goal of obtaining a basis by computational routines, using the language of differential polynomials. We use Wilson's results on weigheted ODOs to guarantee the solvability of the triangular system that allows to compute the homogeneous almost commuting operator of a given order in the ring of ODOs. As a consequence the computation of the equations of the GD hierarchies is obtained without using pseudo-differential operators. The algorithms to calculate the almost commuting basis and the GD hierarchies in the ring of ODOs are implemented in SageMath, and explicit examples are provided.