Painleve equations, vector fields, and ranks in differential fields

Abstract

Model theoretic ranks of solutions to Painleve equations are calculated, and the type of the generic solution of the second Painleve equation is shown to be disintegrated, strengthening a theorem of Nagloo. A question of Hrushovski and Scanlon regarding Lascar rank and Morley rank in differential fields is answered using planar vector fields.

7 pages