First order algebraic differential equations of genus zero

TL;DR

This paper characterizes the transcendental solutions of algebraic first order differential equations of genus zero using recent advances in Riccati equations, providing a detailed understanding of their nature.

Contribution

It offers a comprehensive description of transcendental solutions for genus zero algebraic first order differential equations, building on recent Riccati equation results.

Findings

Transcendental solutions are thoroughly characterized.

The nature of solutions is clarified for genus zero equations.

New connections to Riccati equations are established.

Abstract

We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus zero.