Internality of autonomous algebraic differential equations

TL;DR

This paper investigates when solutions of autonomous algebraic differential equations can be expressed as rational functions of fixed solutions and constants, providing a comprehensive classification for a broad class of systems.

Contribution

It fully characterizes the internality of solutions for a large class of autonomous algebraic differential systems, extending previous results and applying to notable equations.

Findings

Complete classification for a large class of systems.

Necessary condition for solutions to be Liouvillian.

Solutions to Lotka-Volterra are almost never Liouvillian.

Abstract

This article is interested in internality to the constants of systems of autonomous algebraic ordinary differential equations. Roughly, this means determining when can all solutions of such a system be written as a rational function of finitely many fixed solutions (and their derivatives) and finitely many constants. If the system is a single order one equation, the answer was given in an old article of Rosenlicht. In the present work, we completely answer this question for a large class of systems. As a corollary, we obtain a necessary condition for the generic solution to be Liouvillian. We then apply these results to determine exactly when solutions to Poizat equations (a special case of Li\'enard equations) are internal, answering a question of Freitag, Jaoui, Marker and Nagloo, and to the classic Lotka-Volterra system, showing that its generic solutions are almost never Liouvillian.