On the algebraic independence of generic Painleve transcendents

Abstract

We prove that if y" = f(y,y',t) is a generic Painleve equation from among the classes II to V then any collection of distinct solutions and their derivatives are algebraically independent over C(t). (Already proved by Nishioka for the single Painleve I equation). For generic Painleve VI we prove a slightly weaker statement.

11 pages

Cited by in corpus (8)

• When any three solutions are independent
• Ax-Schanuel and strong minimality for the $j$-function
• Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields
• Geometric Triviality of the Strongly Minimal Second Painlevé equations
• Algebraic Independence of generic Painlevé Transcendents: P_III and P_VI
• On Transformations in the Painlevé Family
• Internality of logarithmic-differential pullbacks
• Transcendental numbers as solutions to arithmetic differential equations