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arXiv · 1802.01023

A symplectic Kovacic's algorithm in dimension 4

Thierry Combot·Camilo Sanabria
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Abstract

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.

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Thierry Combot, Camilo Sanabria. 2018-02-03. A symplectic Kovacic's algorithm in dimension 4. https://arxiv.org/abs/1802.01023

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