微分算子的有效下降
Effective descent for differential operators
摘要：
的一个定理。Katz \cite{Ka} p.45指出，在适当的微分域$k$上的不可约微分算子$L$，在$k$的代数闭包上具有同构分解，是$k$上的绝对不可约算子$M$和具有有限微分伽罗瓦群的$k$上的不可约算子$N$的张量积$L=M\otimes_k N$。利用张量分解$L=M\otimes N$的存在性，在\cite{C-W}中给出了一个算法，该算法计算$k$的有限扩张上的$L$的绝对不可约因子$F$.在这里，算法的方法来寻找$M$和$N$的基础上，$F$的知识。这涉及到一个微妙的下降问题的微分算子，可以解决显式微分场$k$这是$C_1$ -领域。
英文摘要：
A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields.
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