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Centro de Ciencias Matemáticas UNAM

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Complex Polynomials and Tessellations in the Riemann Sphere

Abstract:

Let P be a complex polynomial of degree in the Riemann sphere, and let be an oriented Jordan path running through its critical values. A classical algorithm, rooted in the pioneering work of H. A. Schwarz and F. Klein, states that the inverse image of under P determines a finite tessellation of the Riemann sphere, with tiles that are topological k–polygons and alternate colors. Following a question by W. P. Thurston, we study under what conditions a finite graph (or equivalently a finite tessellation of the Riemann sphere) originates from a generic polynomial P and an oriented Jordan path as above.
Journal: Results in Mathematics
ISSN: 1420-9012
Year: 2025
Volume: 80
Pages: 166
Created Created: 2025-08-19 18:27:04
Modified Modified: 2025-08-19 18:27:26
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