Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On dessins d'enfants with equal supports

Fedor Pakovich

TL;DR

We address when dessins on the Riemann sphere have equal supports, defined by $supp D_{\beta}=\beta^{-1}([-1,1])$. The approach embeds these supports into rational lemniscates $L_{\beta}$ and employs results on lemniscate intersections to derive rigidity when $C(\beta_1,\beta_2)=\mathbb{C}(z)$ and $supp D_{\beta_1}=supp D_{\beta_2}$; in this case, the two dessins must arise from either segments or circles. The main theorem shows that under these conditions the dessins are either both segments or both circles; coprimality of degrees via Lüroth ensures the central hypothesis in many cases. The work links the geometry of lemniscates to the combinatorics of dessins, clarifying when different Belyi functions can share equal supports and highlighting explicit families such as Chebyshev polynomials (segments) and unit-circle parametrizations (circles).

Abstract

For a Belyi function $β:\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ ramified only over the points $-1,1,\infty$, a corresponding ``dessin d'enfant'' $\mathcal D_β$ is defined as the set $β^{-1}([-1,1])$ considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets $β^{-1}\{-1\}$ and $β^{-1}\{1\}$ correspondingly. Merely the set $β^{-1}([-1,1])$ without a graph structure is called a support of $\mathcal D_β$. In this note, we solve the following problem: under what conditions different dessins $\mathcal D_{β_1}$ and $\mathcal D_{β_2}$ have equal supports?

On dessins d'enfants with equal supports

TL;DR

We address when dessins on the Riemann sphere have equal supports, defined by . The approach embeds these supports into rational lemniscates and employs results on lemniscate intersections to derive rigidity when and ; in this case, the two dessins must arise from either segments or circles. The main theorem shows that under these conditions the dessins are either both segments or both circles; coprimality of degrees via Lüroth ensures the central hypothesis in many cases. The work links the geometry of lemniscates to the combinatorics of dessins, clarifying when different Belyi functions can share equal supports and highlighting explicit families such as Chebyshev polynomials (segments) and unit-circle parametrizations (circles).

Abstract

For a Belyi function ramified only over the points , a corresponding ``dessin d'enfant'' is defined as the set considered as a bi-colored graph on the Riemann sphere whose white and black vertices are points of the sets and correspondingly. Merely the set without a graph structure is called a support of . In this note, we solve the following problem: under what conditions different dessins and have equal supports?
Paper Structure (4 sections, 3 theorems, 25 equations)

This paper contains 4 sections, 3 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Lemniscates and dessins d'enfants
  3. Intersections of lemniscates
  4. Proof of Theorem \ref{['t1']}

Key Result

Theorem 1.1

Let $\EuScript D_{\beta_1}$ and $\EuScript D_{\beta_2}$ be dessins such that ${\rm supp\,}\{\EuScript D_{\beta_1}\}={\rm supp\,}\{\EuScript D_{\beta_2}\}$ and ${\mathbb C}(\beta_1,\beta_2)={\mathbb C}(z)$. Then either $\EuScript D_{\beta_1}$ and $\EuScript D_{\beta_2}$ are segments, or $\EuScript D_

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3