# A Symplectic Kovacic's Algorithm in Dimension 4

Abstract : Let L be a 4th order linear differential operator with coefficients in K(z), with 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 Xt 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.
Document type :
Conference papers

https://hal.archives-ouvertes.fr/hal-01997563
Contributor : Imb - Université de Bourgogne <>
Submitted on : Tuesday, January 29, 2019 - 10:20:22 AM
Last modification on : Wednesday, January 30, 2019 - 1:26:10 AM

### Citation

Thierry Combot, Camilo Sanabria. A Symplectic Kovacic's Algorithm in Dimension 4. ISSAC 2018, Jul 2018, New York, United States. pp.143-150, ⟨10.1145/3208976.3209005⟩. ⟨hal-01997563⟩

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