Unitary monodromy implies the smoothness along the real axis for some Painlev\'{e} VI equation, I

TL;DR

This paper investigates specific solutions to a particular Painlevé VI equation, establishing explicit pole counts, identifying pole-free solutions, and linking unitary monodromy to smoothness along the real axis.

Contribution

It provides explicit formulas for pole counting, classifies pole-free solutions, and connects monodromy properties to the smoothness of solutions on the real line.

Findings

Explicit pole count formula for algebraic solutions with dihedral monodromy

Only four pole-free solutions exist outside {0,1}

Unitary monodromy implies solutions are smooth on the real axis

Abstract

In this paper, we study the Painlev\'{e} VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left \{ 0,1\right \} $. (iii) If the monodromy group of the associated linear ODE of a solution $\lambda \left( t\right) $ is unitary, then $\lambda ( t) $ has no poles in $\mathbb{R}% \backslash \{ 0,1\} $.