# Autonomous first order differential equations

## TL;DR

This paper provides a geometric classification of solutions to autonomous first order differential equations over algebraically closed fields, using curve geometry and Jacobians, and explores implications for the finiteness properties of solutions.

## Contribution

It offers a complete algebraic and geometric classification of solutions, independent of model-theoretic methods, and addresses the D^n-finiteness of solutions.

## Key findings

- Complete classification of solutions based on curve geometry
- Application to D^n-finiteness of solutions
- Use of generalized Jacobians for analysis

## Abstract

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on $D^n$-finiteness of solutions of first order differential equations.