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Internality of autonomous algebraic differential equations

Christine Eagles, Léo Jimenez

TL;DR

The article develops a model-theoretic criterion for internality to the constants for autonomous algebraic ODEs, linking internality to the Lie-derivative structure of rational first integrals along the system's vector field $X$. By identifying the binding group as the $\mathcal{C}$-points of a linear algebraic group and employing logarithmic differential equations on basic groups, it yields an explicit condition: a set of algebraically independent rational functions $g_i$ with $\mathcal{L}_X(g_i) \in \{\lambda_i g_i, 1\}$ (with at most one $g_i$ achieving 1) characterizes almost internality and weak orthogonality. This leads to concrete classifications for Poizat equations and Lotka–Volterra systems, showing almost internality only in specific structural cases (e.g., Poizat with constant $f$, Lotka–Volterra with $a=c$ up to analyticity), and clarifies when generic solutions are not Liouvillian. The work also discusses pullbacks and generalizations, highlighting when Rosenlicht-type criteria extend and outlining future directions for non-autonomous or variety-based differential systems. Overall, the paper provides a practical, mathematically grounded framework to determine when autonomous algebraic ODEs admit a rich internal structure controlled by constants.

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.

Internality of autonomous algebraic differential equations

TL;DR

The article develops a model-theoretic criterion for internality to the constants for autonomous algebraic ODEs, linking internality to the Lie-derivative structure of rational first integrals along the system's vector field . By identifying the binding group as the -points of a linear algebraic group and employing logarithmic differential equations on basic groups, it yields an explicit condition: a set of algebraically independent rational functions with (with at most one achieving 1) characterizes almost internality and weak orthogonality. This leads to concrete classifications for Poizat equations and Lotka–Volterra systems, showing almost internality only in specific structural cases (e.g., Poizat with constant , Lotka–Volterra with up to analyticity), and clarifies when generic solutions are not Liouvillian. The work also discusses pullbacks and generalizations, highlighting when Rosenlicht-type criteria extend and outlining future directions for non-autonomous or variety-based differential systems. Overall, the paper provides a practical, mathematically grounded framework to determine when autonomous algebraic ODEs admit a rich internal structure controlled by constants.

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.
Paper Structure (12 sections, 25 theorems, 69 equations)

This paper contains 12 sections, 25 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model theory
  4. Logarithmic-differential equations
  5. Criteria for internality
  6. Weakly orthogonal case
  7. Non-weakly orthogonal case
  8. Applications
  9. Poizat equations
  10. Lotka-Volterra equations
  11. Pullbacks
  12. Remarks on generalizations

Key Result

Theorem 1

Work in a saturated model $\mathcal{U} \models \mathrm{DCF}_0$. Let $F$ be a field of constants and $f \in F(x)$. Then the generic type of $y' = f(y)$ is almost internal to the constants if and only if either $f = 0$, or $\frac{\partial g}{\partial x} f = 1$, or $\frac{\partial g}{\partial x}f = cg$

Theorems & Definitions (56)

  • Theorem : Rosenlicht
  • Theorem : In $\mathrm{DCF}_0$
  • Corollary
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.5
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.11
  • ...and 46 more