Computation of RS-pullback transformations for algebraic Painleve VI solutions

TL;DR

This paper develops methods to compute explicit RS-pullback transformations for algebraic solutions of the Painleve VI equation, enabling the derivation of solutions from rational coverings without complex hypergeometric computations.

Contribution

It introduces a systematic approach for deriving algebraic Painleve VI solutions via RS-pullback transformations from rational coverings, simplifying previous methods.

Findings

Explicit RS-pullback transformations computed

Algebraic Painleve VI solutions derived from coverings

Different solutions obtained from the same covering

Abstract

Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked Fuchsian equation. But the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. This paper presents computations of explicit RS-pullback transformations, and derivation of algebraic Painleve VI solutions from them.