Singular Degree of a Rational Matrix Pseudodifferential Operator
Singular Degree of a Rational Matrix Pseudodifferential Operator
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有理矩阵伪微分算子的奇异次
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
V. Kac
中科院分区：
文献类型：
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作者：
Sylvain Carpentier;A. Sole;V. Kac
在我们以前的工作中，我们研究了有理矩阵伪微分算子H=A/B的极小分式分解，其中A和B是矩阵微分算子，B是最小可能度deg（B）的非退化算子.本文引入奇异度sdeg（H）=deg（B），证明了对任意有理表达式H=sum_a（A^a_1）/（B^a_1）...（A^a_n）/（B^a_n），我们得到sdeg（H）小于或等于sum_{a，i} deg（B^a_i）。如果等式成立，我们称这样的表达式为极小表达式。我们研究了奇异度和极小有理表达式的性质。这些结果对于Lenard-Magri可积性方案的计算是很重要的。
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.