A symplectic Kovacic's algorithm in dimension 4

TL;DR

This paper introduces an algorithm to identify and solve projectively symplectic fourth-order differential operators, leveraging Kovacic's algorithm and Klein's Theorem to find Liouvillian and algebraic solutions.

Contribution

It develops the first algorithm to test for projectively symplectic operators and compute their Liouvillian solutions in dimension four.

Findings

Algorithm successfully tests for projectively symplectic operators.

Computes Liouvillian solutions using Kovacic's algorithm.

Provides algebraic solutions as pullbacks of hypergeometric equations.

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.