Remarks Towards the Classification of $RS_4^2(3)$-Transformations and Algebraic Solutions of the Sixth Painlev\'e Equation

TL;DR

This paper introduces a divisor type concept for rational functions to classify deformations of dessins d'enfants, aiding in constructing algebraic solutions to the sixth Painlevé equation through RS-transformations.

Contribution

It presents a novel classification method using divisor types for rational functions, enhancing the understanding of algebraic solutions of Painlevé VI.

Findings

Effective classification of deformations of dessins d'enfants

Construction of algebraic solutions to Painlevé VI

New insights into RS-transformations

Abstract

We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d'enfants related with the construction of the algebraic solutions of the sixth Painlev\'e equation via the method of $RS$-transformations.