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**Geometric properties of the Riemann surfaces associated with the Noumi-Yamada systems with a large parameter.**
*(English)*
Zbl 1242.34154

The Noumi-Yamada systems \((NY)_l\), \(l=2,3,4,\dots\), are systems of \(l+1\) first order ODEs for \(l+1\) unknowns which enjoy the affine Weyl group symmetry of type \(A_l^{(1)}\). These systems can be understood as the generalizations of the classical Painlevé equations PII, PIV and PV. Using appropriate scaling, it is possible to introduce the large parameter \(\eta\) and to study particular \(0\)-parameter solutions to \((NY)_l\) using the so-called exact WKB analysis.

The above mentioned \(0\)-parameter formal series in \(\eta^{-1}\) is completely specified by the leading order term satisfying a system of algebraic equations. Earlier, the authors proved the existence of a \(0\)-parameter solution in the case of even \(l\). In the present paper, they treat the cases of even and odd \(l\) in a unified manner proving that the relevant algebraic system determines a variety which is isomorphic to the variety determined by a hyperbolic system with an appropriate normalization. In this way, the authors find the number of the possible leading order terms (i.e. the number of the branches of a multi-valued holomorphic function satisfying the algebraic system) outside the set of the turning points of the first kind (zeros of the Jacobian of the algebraic system) as well as the number of such turning points.

The above mentioned \(0\)-parameter formal series in \(\eta^{-1}\) is completely specified by the leading order term satisfying a system of algebraic equations. Earlier, the authors proved the existence of a \(0\)-parameter solution in the case of even \(l\). In the present paper, they treat the cases of even and odd \(l\) in a unified manner proving that the relevant algebraic system determines a variety which is isomorphic to the variety determined by a hyperbolic system with an appropriate normalization. In this way, the authors find the number of the possible leading order terms (i.e. the number of the branches of a multi-valued holomorphic function satisfying the algebraic system) outside the set of the turning points of the first kind (zeros of the Jacobian of the algebraic system) as well as the number of such turning points.

Reviewer: Andrei A. Kapaev (Trieste)

### MSC:

34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |

34M60 | Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) |

34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |

34M25 | Formal solutions and transform techniques for ordinary differential equations in the complex domain |

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\textit{T. Aoki} and \textit{N. Honda}, J. Math. Soc. Japan 63, No. 4, 1085--1119 (2011; Zbl 1242.34154)

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### References:

[1] | T. Aoki and N. Honda, Regular sequence associated with the Noumi-Yamada equations with a large parameter, Algebraic Analysis of Differential Equations, Springer, 2008, pp.,45-53. · Zbl 1162.34072 |

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[3] | T. Suwa, Dual class of a subvariety, Tokyo J. Math., 23 (2000), 51-68. · Zbl 0971.32005 |

[4] | Y. Takei, Toward the exact WKB analysis for higher-order Painlevé equations - The case of Noumi-Yamada Systems -, Publ. Res. Inst. Math. Sci., 40 (2004), 709-730. · Zbl 1076.34099 |

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[6] | M. Noumi and Y. Yamada, Symmetry in Painlevé equations, In: Toward the Exact WKB Analysis of Differential Equations, (eds. C. J. Howls, T. Kawai and Y. Takei), Linear or Non-Linear, Kyoto Univ. Press, 2000, pp.,245-260. |

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