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The Painlevé equations were discovered as nonlinear ordinary differential equations that define new special functions, and their importance has long been recognized. Since the 1990s, there have been many studies on various generalizations of the Painlevé equations such as discretizations, higher dimensional analogues, quantizations, and so on. The aim of this book is to provide a unified approach to understand higher dimensional analogues of the Painlevé equations from the viewpoint of the deformation theory of linear ordinary differential equations. Especially, a detailed study will be given when the phase spaces of their Hamiltonian systems are four dimensional. More specifically, starting from the classification of the Fuchsian equations with four accessory parameters, we construct a degeneration scheme of linear equations by considering confluences of singular points. Then we write down the Hamiltonians of the Painlevé-type equations associated with these resulting linear equations. The following topics are explained together with examples: spectral types of linear equations, a method to calculate the Hamiltonians, confluences of singularities and degenerations of the Painlevé-type equations, the correspondence between linear equations or their spectral types through the Laplace transform. In addition, Appendix 1 discusses symmetries of moduli spaces of linear equations. As its application, it is shown that the equations obtained in this book constitute a complete list of 4-dimensional Painlevé-type equations corresponding to unramified linear equations. Appendix 2 gives a list of the 4-dimensional Painlevé-type equations corresponding to ramified linear equations.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
