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 ( 9 8 , − 1 8 , 1 8 , 3 8 ) . We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group of the associated linear ODE being D N , where D N is the dihedral group of order 2 N . (ii) There are only four solutions without poles in C ∖ { 0 , 1 } . (iii) If the monodromy group of the associated linear ODE of a solution λ ( t ) is unitary, then λ ( t ) has no poles in R ∖ { 0 , 1 } .