2024/03/17 by Jaume Alonso, Alonso, Jaume, Yuri B. Suris +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2403.11349
openalex publication_date 2024/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Discrete Painlevé equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlevé equation as a birational map between generalized Halphen surfaces (surfaces obtained from \mathbb P1×\mathbb P1 by blowing up at eight points). We propose a novel geometric interpretation of discrete Painlevé equations, where the family of generalized Halphen surfaces is replaced by a pencil of quadrics in \mathbb P3. A discrete Painlevé equation is viewed as an autonomous birational transformation of \mathbb P3 that preserves the pencil and maps each quadric of the pencil to a different one, according to a Möbius transformation of the pencil parameter. Thus, our scheme is based on the classification of pencils of quadrics in \mathbb P3.