Abstract
Common numerical methods represent Riemann surfaces through algebraic curves, see Chaps. 2 and 3. This seems natural because algebraic curves can be used to define a Riemann surface, but this approach has some serious disadvantages: if one is interested in the corresponding Riemann surface only, one needs to factorize algebraic curves with respect to birational maps. This complicates the corresponding parameterization of Riemann surfaces. Moreover, a representation of a Riemann surfaces as a ramified multi-sheeted covering complicates the description of homology and integration paths, which leads to complex algorithms. Schottky uniformization (see Chap. 1) is attractive alternative to describing Riemann surfaces in terms of algebraic curves.
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Schmies, M. (2011). Computing Poincaré Theta Series for Schottky Groups. In: Bobenko, A., Klein, C. (eds) Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics(), vol 2013. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17413-1_4
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DOI: https://doi.org/10.1007/978-3-642-17413-1_4
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