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arXiv · 1801.08529

Fuchsian Equations with Three Non-Apparent Singularities

Alexandre Eremenko·Vitaly Tarasov
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Abstract

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

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Alexandre Eremenko, Vitaly Tarasov. 2018-01-25. Fuchsian Equations with Three Non-Apparent Singularities. https://doi.org/10.3842/sigma.2018.058

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