### Abstract

In this paper, we study the Painlevé VI equation with parameter (98,−18, 18, 38). 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 2N. (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}.

Original language | English |
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Pages (from-to) | 52-63 |

Number of pages | 12 |

Journal | Journal of Geometry and Physics |

Volume | 116 |

DOIs | |

Publication status | Published - 2017 Jun 1 |

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### Keywords

- Algebraic solution
- Painlevé VI equation
- Pole distribution

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

### Cite this

*Journal of Geometry and Physics*,

*116*, 52-63. https://doi.org/10.1016/j.geomphys.2017.01.016

**Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I.** / Chen, Zhijie; Kuo, Ting-Jung; Lin, Chang Shou.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 116, pp. 52-63. https://doi.org/10.1016/j.geomphys.2017.01.016

}

TY - JOUR

T1 - Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I

AU - Chen, Zhijie

AU - Kuo, Ting-Jung

AU - Lin, Chang Shou

PY - 2017/6/1

Y1 - 2017/6/1

N2 - In this paper, we study the Painlevé VI equation with parameter (98,−18, 18, 38). 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 DN, where DN is the dihedral group of order 2N. (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}.

AB - In this paper, we study the Painlevé VI equation with parameter (98,−18, 18, 38). 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 DN, where DN is the dihedral group of order 2N. (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}.

KW - Algebraic solution

KW - Painlevé VI equation

KW - Pole distribution

UR - http://www.scopus.com/inward/record.url?scp=85011632800&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85011632800&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2017.01.016

DO - 10.1016/j.geomphys.2017.01.016

M3 - Article

AN - SCOPUS:85011632800

VL - 116

SP - 52

EP - 63

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -