Classifying (almost)-Belyi maps with Five Exceptional Points

TL;DR

This paper classifies specific rational functions with five exceptional points related to hypergeometric functions, extending previous work on four exceptional points to aid in solving second order linear differential equations.

Contribution

It provides a complete classification of rational functions with five exceptional points, advancing the understanding of their structure and applications in differential equations.

Findings

Classified all rational functions with five exceptional points.

Extended previous classification for four points to five points.

Facilitates solving second order linear differential equations with five true singularities.

Abstract

We classify all rational functions whose branching pattern above {0, 1, infinity} satisfy a certain regularity condition with precisely d=5 exceptions. This work is motivated by solving second order linear differential equations, with d=5 true singularities, in terms of hypergeometric functions. A similar problem was solved for d=4 by Vidunas and Filipuk.