Any six points on the Riemann sphere can be split into three pairs by a triple of disjoint discs

TL;DR

The paper proves that six points on the Riemann sphere can be separated into three pairs by disjoint discs, enabling a numerical method for evaluating genus 2 hyperelliptic functions.

Contribution

It establishes a geometric partitioning result that supports a new numerical approach for hyperelliptic functions of genus 2.

Findings

Six points can be separated into three pairs by disjoint discs.

This geometric result validates a numerical method for genus 2 hyperelliptic functions.

The method applies to any complex curve of genus 2.

Abstract

We prove that for any six points on the Riemann sphere there exist three disjoint closed (or open) discs, each of which contains exactly two of the six distinguished points. This statement shows that recently proposed method to numerically evaluate Kleinian hyperelliptic functions of genus 2 is applicable to any complex curve of genus 2.