Singular Degree of a Rational Matrix Pseudodifferential Operator
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Author(s)
Carpentier, Sylvain
De Sole, Alberto
Kac, Victor
Date Issued
June 2014
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Citation
Carpentier, Sylvain, et al. “Singular Degree of a Rational Matrix Pseudodifferential Operator.” International Mathematics Research Notices, vol. 2015, no. 13, 2015, pp. 5162–95. © 2014 The Authors
Version
Original manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNU093
