Internality of autonomous algebraic differential equations
arXiv:2409.01863
The paper investigates when solutions of autonomous algebraic ordinary differential equations can be written as rational functions of finitely many fixed solutions (and their derivatives) together with constants, providing a complete characterization for a large class of systems and applying the results to Poizat equations and the Lotka‑Volterra system.
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 Liénard 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.
Many improvements were made to the article. Many thanks to our anonymous referees for their suggestions