Singular Degree of a Rational Matrix Pseudodifferential Operator

Abstract

In our previous work, we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H = AB[superscript -1], where Aand B are matrix differential operators, and B is nondegenerate 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 =∑[subscript α] A[subscript 1][superscript ] (B[subscript 1][superscript α])[superscript -1] ⋯ A[subscript n][superscript α] (B[subscript n][superscript α])[superscript -1], we have sdeg(H) ≤∑[subscript α,i] deg(B[subscript i][superscript α]). 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.