An algorithm for construction of commuting ordinary differential operators by geometric data
The aim of this paper is to give a description and program realization of an algorithm which produces explicit examples of commuting ordinary differential operators with rational coefficients by geometric spectral data consisting of a rational curve and a torsion free sheaf with some trivialization. The paper is organized as follows. The first section is an introduction to the subject. In Section 2 the authors give a survey on Schur pairs, geometric data and commuting differential operators to explain all theoretical ingredients they use in the algorithm and to make the paper self-contained. Section 3 contains the explanation of the first part of the algorithm. In particular, in this section they explain two ways of calculation Schur pairs from spectral data. The major new result is about direct images of torsion free sheaves (Any torsion free sheaf over a rational curve is the direct image of a torsion free rank one sheaf over another rational curve). The section also contains several examples of explicit Schur pairs. Section 4 contains a description of the algorithm. In Section 5 the authors give several test examples of computer calculations. In Appendix they give an overview of codes and procedures together with an estimate of their complexity.
commuting ordinary differential operators
scientific article; zbMATH DE number 6846092
scientific article