On dessins d'enfants with equal supports

2025/05/18 by Fedor Pakovich, Pakovich, Fedor · 1 citation
Mathematics · #Holomorphic and Operator Theory #Analytic and geometric function theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2505.12420

Abstract

For a Belyi function β:\mathbb C\mathbb P¹→ \mathbb C\mathbb P¹ ramified only over the points -1,1,∞, 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_β1 and \mathcal D_β2 have equal supports?

Citations

• On the Shapes of Rational Lemniscates
• On intersection of lemniscates of rational functions
• Level curves of rational functions and unimodular points on rational\n curves
• Shapes, fingerprints and rational lemniscates

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• Holomorphic maps sharing preimages over finitely generated fields

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