# Internality of logarithmic-differential pullbacks

## TL;DR

This paper provides a criterion based on model theory for when certain planar vector fields admit algebraically independent first integrals, focusing on the internality of logarithmic-differential pullbacks in differential fields.

## Contribution

It introduces a model-theoretic criterion for the internality of logarithmic-differential pullbacks, connecting differential algebra and model theory in the context of integrability.

## Key findings

- Criterion for internality of pullbacks in differential fields
- Connection between algebraic independence and model-theoretic internality
- Application to planar vector fields with first integrals

## Abstract

A criterion in the spirit of Rosenlicht is given, on the rational function f(x), for when the planar vector field defined by x'=f(x) and y'=xy admits a pair of algebraically independent first integrals over some extension of the base field. This proceeds from model-theoretic considerations by working in the theory of differentially closed fields of characteristic zero and asking: If D is a strongly minimal set on the affine line that is internal to the constants, when is the pullback of D under the logarithmic derivative itself internal to the constants?