Abstract
In this paper, we try to answer some questions raised by Cangelmi (Eur J Comb 33(7):1444–1448, 2012). We reinterpret the Riemann–Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann–Roch theorems for orientable hypermaps by defining the divisor of a function f on darts. In addition, we extend Riemann–Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann–Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory.
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Acknowledgements
The authors are very grateful to the anonymous reviewer for giving extremely valuable suggestions and comments to improve this paper. This work was partially supported by the National Natural Science Foundation of China (No. 11871260).
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Cheng, M., Cao, T. Riemann–Hurwitz theorem and Riemann–Roch theorem for hypermaps. J Algebr Comb 59, 95–110 (2024). https://doi.org/10.1007/s10801-023-01285-9
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DOI: https://doi.org/10.1007/s10801-023-01285-9