Opuscula Math. 38, no. 2 (2018), 201-252
https://doi.org/10.7494/OpMath.2018.38.2.201

Opuscula Mathematica

Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation

Mitsuo Kato
Toshiyuki Mano
Jiro Sekiguchi

Abstract. A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painlevé VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.

Keywords: flat structure, Painlevé VI equation, algebraic solution, potential vector field.

Mathematics Subject Classification: 34M56, 33E17, 35N10, 32S25.

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• Mitsuo Kato
• University of the Ryukyus, Colledge of Educations, Department of Mathematics, Nishihara-cho, Okinawa 903-0213, Japan
• Toshiyuki Mano
• University of the Ryukyus, Faculty of Science, Department of Mathematical Sciences, Nishihara-cho, Okinawa 903-0213, Japan
• Jiro Sekiguchi
• Tokyo University of Agriculture and Technology, Faculty of Engineering, Department of Mathematics, Koganei, Tokyo 184-8588, Japan
• Communicated by Yoshishige Haraoka.
• Revised: 2017-10-25.
• Accepted: 2017-11-08.
• Published online: 2017-12-29.