Effective computation of centralizers of ODOs

TL;DR

This paper presents an algorithm for computing the centralizer of an ordinary differential operator, linking algebraic and geometric methods, and generating families of operators with non-trivial centralizers.

Contribution

It introduces a novel algorithm combining algebraic and integrable systems techniques to compute centralizers and generate families of operators with non-trivial centralizers.

Findings

Algorithm computes a basis of the centralizer as a module over the polynomial ring.

Method links centralizers to spectral curves and solutions of the stationary Gelfand-Dickey hierarchy.

Generates families of operators with non-trivial centralizers using parametric coefficients.

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.