Painleve equations, vector fields, and ranks in differential fields

TL;DR

This paper calculates model theoretic ranks of Painleve equation solutions, showing the generic solution of the second Painleve equation is disintegrated, and addresses a question on ranks in differential fields using planar vector fields.

Contribution

It advances understanding of the model theoretic properties of Painleve equations and resolves a question about ranks in differential fields with new methods.

Findings

Model theoretic ranks of Painleve solutions are computed.

The generic second Painleve solution is shown to be disintegrated.

A question on Lascar and Morley ranks in differential fields is answered.

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.