Factoring Third Order Ordinary Differential Operators over Spectral Curves

TL;DR

This paper investigates the algebraic structure of third order differential operators over spectral curves, explicitly describing their centralizer and providing an algorithm for factorization, including the first example of a non-planar spectral curve.

Contribution

It explicitly characterizes the centralizer of third order algebro-geometric operators and introduces a symbolic factorization algorithm using differential subresultants.

Findings

The spectral curve is a space curve, not necessarily planar.

Explicit description of the centralizer ring structure.

Development of a factorization algorithm for most points on the spectral curve.

Abstract

We consider the classical factorization problem of a third order ordinary differential operator $L-\lambda$, for a spectral parameter $\lambda$. It is assumed that $L$ is an algebro-geometric operator, that it has a nontrivial centralizer, which can be seen as the affine ring of curve, the famous "spectral curve" $\Gamma$. In this work we explicitly describe the ring structure of the centralizer of $L$ and, as a consequence, we prove that $\Gamma$ is a space curve. In this context, the first computed example of a non-planar spectral curve arises, for an operator of this type. Based on the structure of the centralizer, we give a symbolic algorithm, using differential subresultants, to factor $L-\lambda_0$ for all but a finite number of points $P=(\lambda_0 , \mu_0 , \gamma_0)$ of the spectral curve .