Abstract
Any multivalued algebraic function y(x) becomes univalued on the Riemann surface T associated with it. Consequently, if Q(x, y) is a rational function with two variables, then the algebraic function Q(x, y(x)) also becomes univalued on T. The functions of this type form the field of rational functions on the Riemann surface, which is the fundamental algebraic object associated with T.
Keywords
- Rational Function
- Riemann Surface
- Projective Space
- Meromorphic Function
- Algebraic Curve
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B. Riemann, Theorie der Abelschen Functionen. J. Reine Angew. Math. 54, 115–155 (1857). French translation: Théorie des fonctions abéliennes. Dans Œuvres mathématiques de Riemann, transl. L. Laugel (Gauthier-Villars, Paris, 1898), pp. 89–164. Reprinted by J. Gabay, Sceaux, 1990
G. Roch, Ueber die Anzahl der willkurlichen Constanten in algebraischen Functionen. J. Reine Angew. Math. 64, 372–376 (1865)
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Popescu-Pampu, P. (2016). The Riemann–Roch Theorem. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_16
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DOI: https://doi.org/10.1007/978-3-319-42312-8_16
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