二元线性偏微分算子的多重因子分解
Multiple Factorizations of Bivariate Linear Partial Differential Operators
摘要：
本文研究了二元三阶或四阶线性微分算子（LPDO）有几种不同分解的情形。我们证明了一个三阶二元LPDO有一个一阶左因子和一阶右因子使得它们的符号互素当且仅当该算子有一个分解为三个因子的分解，其中左因子恰好是初始左因子，右因子恰好是初始右因子。我们表明，初始的左，右因子的符号是互质的条件是必不可少的，类似的声明“因为它是”是不正确的LPDO的顺序四。然后，我们考虑完全可约LPDO，这是定义为主理想的交集。这样的运算符也可能需要有几个不同的因式分解。考虑到所有可能的情况，由于本文的第一个结果，我们从考虑中排除了其中一些情况。给出了LPDO完全约化的充分条件的显式表达式。
英文摘要：
We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations. We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four. Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also.
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