Computation of the topological type of a real Riemann surface.(English)Zbl 1298.14067

The authors present an algorithm for the computation of the topological type of a real Riemann surface with an antiholomorphic involution $$\tau$$ associated to an algebraic curve.
The topological type comprises genus, number of connected components of the set of fixpoints of $$\tau$$, and whether or not these connected components divide the Riemann surface into one or two connected components.
An algorithm by C. L. Tretkoff and M. D. Tretkoff [in: Contributions to group theory, Contemp. Math. 33, 467–519 (1984; Zbl 0557.30036)] is used to compute a canonical homology basis and the genus of the surface. The topology type is determined by transforming the canonical homology basis into a homology basis whose $$\mathcal{A}$$-cycles are invariant under $$\tau$$.
The authors demonstrate how a Matlab implementation of the algorithm can be applied to various examples of algebraic curves.

MSC:

 14Q05 Computational aspects of algebraic curves 68W30 Symbolic computation and algebraic computation

Zbl 0557.30036
Full Text:

References:

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