2007/04/05 by Katsunori Iwasaki, Iwasaki, Katsunori
Mathematics · Physics and Astronomy · #34M55 #37F10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #math.CA #msc:34M55 #msc:37F10
paper · pdf · doi:10.48550/arxiv.0704.0679
45 pages, 22 figures, 5 tables
arxiv created 2007/04/05 · openalex publication_date 2007/04/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations.