Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory

TL;DR

This paper extends the classical Burchnall-Chaundy theory from scalar differential operators to matrix operators, introducing a new differential elimination tool called the matrix differential resultant to compute BC-polynomials and analyze spectral problems.

Contribution

The work generalizes BC-polynomials to matrix differential operators and develops the matrix differential resultant as a new tool for their computation and analysis.

Findings

Matrix differential resultant has constant coefficients.

Provides necessary and sufficient conditions for spectral solutions.

Explicitly describes isomorphisms between rings of MODOs and algebraic curves.

Abstract

Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute there exists a polynomial $f(\lambda ,\mu)$ with complex coefficients such that $f(P,Q)=0$, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Matrices have entries in a differential field $K$, whose field of constants $C$ is algebraically closed and of zero characteristic. We restrict to the case of order one operators $P$, with invertible leading coefficient. A new differential elimination tool is defined, the matrix differential resultant. It is used to compute the BC-polynomial $f$ of a pair of commuting MODOs and proved to have constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem $PY=\lambda Y \ , \ QY=\mu Y$ to have a solution. Techniques from differential algebra and Picard-Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs $C[P,Q]$ and a finite product of rings of irreducible algebraic curves.