四维空间上的辛Kovacic算法
A symplectic Kovacic's algorithm in dimension 4
摘要：
设$L$是系数在$\mathbb{K}(z)$中的$4$阶微分算子，$\mathbb{K}$是可计算的代数闭域.算子$L$在有理规范变换下称为辛算子，解的基本矩阵$X$满足$X^t J X=J$，其中$J$是标准辛矩阵.当它与辛算子射影等价时，称为射影辛算子。我们设计了一个算法来测试$L$是否是投影辛的。在Kovacic算法的基础上，我们设计了一个计算$4$阶投影辛算子的Liouville解的算法.此外，利用克莱因定理，代数解作为标准超几何方程的拉回。
英文摘要：
Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions $X$ satisfies $X^t J X=J$ where $J$ is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if $L$ is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order $4$. Moreover, using Klein's Theorem, algebraic solutions are given as pullbacks of standard hypergeometric equations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
