Internality of autonomous algebraic differential equations

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.