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The Riemann–Roch Theorem

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Part of the Lecture Notes in Mathematics book series (HISTORYMS,volume 2162)

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

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 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

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  2. G. Roch, Ueber die Anzahl der willkurlichen Constanten in algebraischen Functionen. J. Reine Angew. Math. 64, 372–376 (1865)

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© 2016 Springer International Publishing Switzerland

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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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