A Note on Integrability and Internality in DCF0

TL;DR

This paper explores the connection between algebraic integrability and model-theoretic internality, providing a geometric interpretation and linking internality to the existence of sufficient 'good' first integrals.

Contribution

It offers a new geometric perspective on internality in model theory and its relation to algebraic integrability.

Findings

Internality corresponds to having enough 'good' first integrals.

Provides a geometric account of almost internality.

Establishes a connection between model theory and algebraic integrability.

Abstract

We investigate the relationship between algebraic integrability and the model theoretic notion of internality. Our main result give a geometric account of almost internality and indeed we show that this notion correspond in a reasonable way to having enough "good" first intergrals.