Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I

Abstract

In this paper, we study the Painlevé 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 $\mathbb{R}$.

18 pages, final version, to appear in Journal of Geometry and Physics

References in corpus (1)

• Hamiltonian system for the elliptic form of Painlevé VI equation

Cited by in corpus (2)

• The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence
• The geometry of generalized Lamé equation, I