# Computing the topology of a planar or space hyperelliptic curve

## TL;DR

This paper introduces algorithms to efficiently compute the topology of 2D and 3D hyperelliptic curves by leveraging their relation to simpler planar curves, with implementation and complexity analysis.

## Contribution

The paper presents novel algorithms that use birational mappings to determine hyperelliptic curve topology, including implementation and complexity considerations.

## Key findings

- Algorithms successfully compute hyperelliptic curve topology
- Implementation in Maple demonstrates practical applicability
- Complexity and certification issues are addressed

## Abstract

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.