具有选择微分Galois群的线性微分算子的典范分解
Canonical decomposition of linear differential operators with selected differential Galois groups
摘要：
我们重新考虑一个六阶线性微分算子的解是一个三元有理函数的对角线。它的外平方有一个有理解，表明它有一个选择的微分伽罗瓦群，并且实际上与它的伴随同态。我们得到了两个相应的缠绕给这个同态的伴随。我们表明，这些缠绕也是同态的，他们的伴随，并有一个简单的分解，已经强调在以前的文件中，在订单，两个自伴运营商。从这些结果中，我们推出了一个新的形式的分解的算子为选定的六阶线性微分算子的三个二阶自伴算子。然后，我们推广了以前的分解分解的任意数量的自伴算子相同的奇偶顺序。这产生了一个无限族的线性微分算子同态到他们的伴随，因此，与选定的微分伽罗瓦群。我们证明了这些算子的等价性与这些正则分解是相容的。这些选定的运营商的对称，或外部，平方的合理解决方案是，值得注意的是，看到只依赖于最右边的自伴算子的分解。这些结果和工具，适用于运营商的大订单。例如，可以看到，一个大的（相当庞大的）算子集，与定义卡-丘3-折叠的自反4-多面体相关联，最近由P. Lairez获得，对应于本文详细描述的分解的特定形式。
英文摘要：
We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.
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