Singular degree of a rational matrix pseudodifferential operator

TL;DR

This paper introduces the concept of singular degree for rational matrix pseudodifferential operators and explores its properties, especially in relation to minimal rational expressions, with implications for integrability schemes.

Contribution

It defines the singular degree for such operators and characterizes minimal rational expressions, advancing the understanding of their structure and properties.

Findings

sdeg(H) is less than or equal to sum of degrees of denominators in a rational expression

Equality of sdeg(H) and sum of degrees indicates a minimal expression

Results aid computations in the Lenard-Magri scheme of integrability

Abstract

In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of minimal possible degree deg(B). In the present paper we introduce the singular degree sdeg(H)=deg(B), and show that for an arbitrary rational expression H=sum_a (A^a_1)/(B^a_1)...(A^a_n)/(B^a_n), we have that sdeg(H) is less than or equal to sum_{a,i} deg(B^a_i). If the equality holds, we call such an expression minimal. We study the properties of the singular degree and of minimal rational expressions. These results are important for the computations involved in the Lenard-Magri scheme of integrability.