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.

Singular degree of a rational matrix pseudodifferential operator / DE SOLE, Alberto; V., Kac; S., Carpentier. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - STAMPA. - 13:(2014), pp. 5162-5195. [10.1093/imrn/rnu093]

Singular degree of a rational matrix pseudodifferential operator

DE SOLE, ALBERTO;
2014

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.
2014
singular degree; pseudodifferential operators; rational matrix
01 Pubblicazione su rivista::01a Articolo in rivista
Singular degree of a rational matrix pseudodifferential operator / DE SOLE, Alberto; V., Kac; S., Carpentier. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - STAMPA. - 13:(2014), pp. 5162-5195. [10.1093/imrn/rnu093]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/639414
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