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Analytic, Algebraic and Geometric Aspects of Differential Equations pp 383–398Cite as

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Flat Structures and Algebraic Solutions to Painlevé VI Equation

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Abstract

The aim of this paper is first to formulate the definition of Frobenius manifolds and its generalization. Then we study the algebraic solutions to Painlevé VI obtained by Dubrovin-Mazzocco related with the reflection group of type H 3.

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Notes

  1. 1.

    Sometimes we treat the case where some of w 1, …, w n are equal. In this paper we use the notation \(\partial _{j} = \partial _{x_{j}}\) for the sake of simplicity.

References

  1. P. Boalch, The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, 183–214 (2006)

    MathSciNet  MATH  Google Scholar 

  2. P. Boalch, Towards a nonlinear Schwarz’s list, in The Many Facets of Geometry (Oxford University Press, Oxford, 2010), pp. 210–236

    Book  MATH  Google Scholar 

  3. B. Dubrovin, Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups. Montecatini Terme, 1993, ed. by M. Francoviglia, S. Greco. Lecture Notes in Mathematics, vol. 1620 (Springer, Berlin/Heidelberg, 1996), pp.120–348

    Google Scholar 

  4. B. Dubrovin, M. Mazzocco, Monodromy of certain Painlevé VI transcendents and reflection groups. Invent. Math. 141, 55–147 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Kato, J. Sekiguchi, Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three. Kyushu J. Math. 68, 181–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Kato, T. Mano, J. Sekiguchi, Flat structures without potentials. Revue Roumaine Math. Pures Appl. 60(4), 481–505 (2015)

    MathSciNet  Google Scholar 

  7. M. Kato, T. Mano, J. Sekiguchi, Flat structure on the space of isomonodromic deformations. Preprint arXiv:1511.01608

    Google Scholar 

  8. F.C. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünnften Grade, in Herausgegeben mit einer Einführung und mit Kommentaren von Perer Slodowy (Birkhäuser, Basel/Boston/Berlin, 1993)

    MATH  Google Scholar 

  9. Y.I. Manin, F-manifolds with flat structure and Dubrovin’s duality. Adv. Math. 198, 5–26 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Sabbah, Isomonodromic Deformations and Frobenius Manifolds. An Introduction. Universitext (Springer, London, 2008)

    Google Scholar 

  11. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 265–291 (1980)

    MathSciNet  MATH  Google Scholar 

  12. K. Saito, On a linear structure of the quotient variety by a finite reflexion group. Preprint RIMS-288 (1979); Publ. RIMS Kyoto Univ. 29, 535–579 (1993)

    Google Scholar 

  13. K. Saito, T. Ishibe, Monoids in the fundamental groups of the complement of logarithmic free divisors in C 3. J. Algebra 344, 137–160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Saito, T. Yano, J. Sekiguchi, On a certain generator system of the ring of invariants of a finite reflection group. Commun. Algebra 8, 373–408 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Sekiguchi, A classification of weighted homogeneous Saito free divisors. J. Math. Soc. Jpn. 61, 1071–1095 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by a grant-in-aid from the Japan Society for the Promotion of Science (Grant Numbers 25800082 and 26400111).

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Educations, University of Ryukyus, Nishihara-cho, Okinawa, 903-0213, Japan

    Mitsuo Kato

  2. Faculty of Sciences, Department of Mathematical Sciences, University of Ryukyus, Nishihara-cho, Okinawa, 903-0213, Japan

    Toshiyuki Mano

  3. Faculty of Engineering, Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo, 184-8588, Japan

    Jiro Sekiguchi

Authors
  1. Mitsuo Kato
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  2. Toshiyuki Mano
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  3. Jiro Sekiguchi
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Corresponding author

Correspondence to Jiro Sekiguchi .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Galina Filipuk

  2. Kumamoto University, Kumamoto, Japan

    Yoshishige Haraoka

  3. Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University, Warsaw, Poland

    Sławomir Michalik

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Kato, M., Mano, T., Sekiguchi, J. (2017). Flat Structures and Algebraic Solutions to Painlevé VI Equation. In: Filipuk, G., Haraoka, Y., Michalik, S. (eds) Analytic, Algebraic and Geometric Aspects of Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-52842-7_11

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