Abstract
In this chapter, we discuss the “permissive” part of topological Galois theory. It is based on the following classical results: a simple linear-algebraic part of Picard–Vessiot theory and Frobenius’s theorem (Theorem 6.2). To prove that a Fuchsian equation with a k-solvable monodromy group is solvable by k-quadratures, we also need to use standard Galois theory.
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Notes
- 1.
Note that the functions f and xf are proportional over \(\hat{M}(0,\varepsilon )\); thus we can arrange that the α’s have real parts between 0 and 1.
- 2.
Indeed, a subgroup of index k defines an action of the group on a k-element set such that the subgroup is the stabilizer of some element. The desired normal subgroup of index ≤ k! is the kernel of this action.
- 3.
These forms of solvability are different unless we restrict the coefficients. The same holds for the forms of solvability appearing in items 2, 4, and 5 below.
References
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Khovanskii, A. (2014). Solvability of Fuchsian Equations. In: Topological Galois Theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38871-2_6
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DOI: https://doi.org/10.1007/978-3-642-38871-2_6
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