Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

By H. Sakai. 2001

633 citations recorded by the source corpus.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/011fb6596304e436805ec665910077521e400594

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Eric M. Rains
2. V. Spiridonov
3. N. Hitchin
4. Walter Van Assche
5. Peter A. Clarkson
6. Y. Yamada
7. Kenji Kajiwara
8. M. Noumi

In Statistical and Nonlinear Physics

1. V. Spiridonov
2. Eric M. Rains
3. N. Hitchin
4. Walter Van Assche
5. Peter A. Clarkson
6. Teruhisa Tsuda
7. M. Noumi
8. Y. Yamada

Citations in the corpus

• Geometric aspects of Painlevé equations (2015, methodology): “[113] H. Sakai, Lax forms of the q-Painlevé equation associated with the A(1)2 surface, J. Phys.”
• Transformations of elliptic hypergometric integrals (2003, result): “This gives rise to a system of three-term quadratic recurrences, which turn out to be a form of Sakai’s elliptic Painlevé equation [29]; this generalizes the result of [13] for the univariate case (the authors of which also observed the existence of an E7 symmetry in that case).”
• Essays on the theory of elliptic hypergeometric functions (2008, methodology): “In [111], Sakai gave a classification of discrete Painlevé equations connected with the affine Weyl groups.”
• Application of the τ-function theory of Painlevé equations to random matrices: PVI , the JUE, CyUE, cJUE and scaled limits (2002, result): “We thus establish directly that the discrete dPV is the contiguity relation of the continuous PVI equation in contrast to other treatments [61], [60], [33], [45].”
• Deformations of holomorphic Poisson manifolds (2011, methodology): “This family of Poisson surfaces arises in [20] in the context of the first Painlevé equation y = 6y(2) + x.”
• Painlevé equations: nonlinear special functions (2003, background): “1) is so named since it is of type D7 in the terminology of Sakai [202].”
• How instanton combinatorics solves Painlevé VI, V and IIIs (2013, background): “Significance of the degenerate equations P III2 and P III3 was realized in [49] from a geometric viewpoint, and later they were extensively studied in [46].”
• Discrete Painlev\'e equations for recurrence coefficients of orthogonal polynomials (2005, background): “Sakai [21] made a classification in terms of rational surfaces associated with affine root systems and the most general (elliptic) discrete Painlevé equation is related with the affine Weyl group symmetry of type E8.”
• Folding transformations of the Painlevé equations (2005, methodology): “Folding transformations of the Painlev´ e equations 719 Halphen surface ([ 12 ]).”