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I was reading Kovacic's algorithm for solving second-order linear homogeneous ODEs with coefficients in C(x) and wondering about extending it to handle algebraic functions as coefficients.

The paper ("An algorithm for solving second order linear homogeneous differential equations") uses linear algebraic groups extensively, and I'm wondering how to handle them over C(x) instead of C. This may (or may not) be needed to extend the algorithm.

So what do we know about GL(2,C(x)), for example? Or SL(2,C(x))?

What are these groups even called? I can't seem to Google for the right name, but it's hard to believe that they haven't been studied.

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