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

TL;DR

The paper addresses the problem of algebraic independence among solutions of the classical Lotka-Volterra system by establishing strong minimality in a universal differential field when $d$ and $b$ are linearly independent over $\\mathbb{Q}$. The authors combine Puiseux-series techniques with results on strong minimality and geometric triviality to show that distinct non-constant solutions are algebraically independent over $\\mathbb{C}$, producing $tr.deg_{\\mathbb{C}}\\mathbb{C}(x_1,y_1,...,x_n,y_n)=2n$. They also obtain partial results for the generalized $2d$-Lotka-Volterra system, proving strong minimality under analogous rational-independence assumptions. The work leverages invariant-curve analysis, Brestovski form, and model-theoretic stability concepts to translate dynamic independence into a robust algebraic-transcendence claim, with implications for ω-categoricity considerations in second-order strongly minimal differential equations.

Abstract

Let $(x_1,y_1),\ldots,(x_n,y_n)$ 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 $tr.deg_{\mathbb{C}}\mathbb{C}(x_1,y_1,\ldots,x_n,y_n)=2n$. 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.

Theorems & Definitions (36)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.9