Computation of the topological type of a real Riemann surface

TL;DR

This paper introduces an algorithm to determine the topological characteristics of real compact Riemann surfaces, including genus and fixed point set properties, by transforming homology bases to reveal involution invariance.

Contribution

The paper presents a novel algorithm for computing the topological type of real Riemann surfaces through homology basis transformation.

Findings

Algorithm effectively computes genus and fixed point set properties.

Transforms homology basis to reveal involution invariance.

Provides a systematic method for topological classification.

Abstract

We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution $\tau$, namely, the number of its connected components, and whether this set divides the surface into one or two connected components. This is achieved by transforming an arbitrary canonical homology basis to a homology basis where the $\mathcal{A}$-cycles are invariant under the anti-holomorphic involution $\tau$.