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

Matvey Smirnov

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.

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

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.

Paper Structure

This paper contains 5 sections, 30 theorems, 89 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Basic reduction of the problem and the idea of the proof
  3. Splitting by a strip
  4. Splitting using discs with diameter on a pair of points from E
  5. Splitting in the remaining cases

Key Result

Theorem 1.1

Let $A \subset \mathop{\mathrm{\mathbb C}}\nolimits\mathop{\mathrm{\mathbb P}}\nolimits(1)$ be any six-element set. Then there exist three disjoint closed discs $D_1,D_2,D_3 \subset \mathop{\mathrm{\mathbb C}}\nolimits\mathop{\mathrm{\mathbb P}}\nolimits(1)$ such that $D_j \cap A$ consists of exactl

Figures (8)

  • Figure 1: The set $\Omega$; the boundary of $\Omega$ is shown by the solid line.
  • Figure 2: An example of a set $E = \{z,w,v\}$ splittable by a strip.
  • Figure 3: The sets $A$ and $B$ from \ref{['eqAandB']}.
  • Figure 4: The disc $F$ from Lemma \ref{['lemDeformedCircleInImpersandCase']}.
  • Figure 5: Example of a pair of approximately collinear points $z_1, z_2 \in \mathop{\mathrm{\mathbb C}}\nolimits \setminus \Omega$ such that $F(z_1,z_2) \cap \overline{\mathop{\mathrm{\mathbb D}}\nolimits} \ne \emptyset$.
  • ...and 3 more figures

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 52 more