Order one equations with the Painlev\'e property

TL;DR

This paper bridges classical and modern approaches to order one differential equations with the Painlevé property, using algebraic geometry and differential Galois theory to provide a comprehensive classification.

Contribution

It connects classical and modern theories of Painlevé equations and introduces a new classification using algebraic geometry and differential Galois theory.

Findings

Unified classical and modern framework for Painlevé equations

New classification scheme for order one equations with Painlevé property

Applicable to both characteristic 0 and p

Abstract

Differential equations with the Painlev\'e property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order one equations appeared before, the connection with the classical theory did not. Using techniques from algebraic geometry we provide the link between the classical and the modern treatment, and with the help of differential Galois theory a new classification is derived, both for characteristic 0 and p.