Effective computation of centralizers of ODOs

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.