Algebraic independence of the solutions of the classical Lotka-Volterra system

Abstract

Let $(x_1,y_1),\dots ,(x_n,y_n)$ be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system \begin{equation}\notag \begin{split} x'&= axy + bx\\ y'&= cxy + dy, \end{split} \end{equation} where $a,b,c,d\in \mathbb{C}\setminus \{0\}$. We show that if $d$ and $b$ are linearly independent over $\mathbb{Q}$, then the solutions are algebraically independent over $\mathbb{C}$, that is $tr.de{g}_{\mathbb{C}}\mathbb{C}(x_1,y_1,\dots ,x_n,y_n)=2n$. As a main part of the proof, we show that the set defined by the system in universal differential fields, with $d$ and $b$ linearly independent over $\mathbb{Q}$, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general $2d$-Lotka-Volterra system.

17 pages. Authors' version to appear in Annals of Pure and Applied Logic