Algebraic independence of solutions to multiple Lotka-Volterra systems

TL;DR

This paper proves that solutions to certain multi-dimensional Lotka-Volterra systems are algebraically independent under specific conditions, extending previous work and classifying invariant algebraic curves in special cases.

Contribution

It establishes algebraic independence of solutions for a broad class of Lotka-Volterra systems and classifies invariant algebraic curves when the systems are not strongly minimal.

Findings

Solutions are algebraically independent when $b_i eq d_i$ and parameters are distinct.

Strong minimality of systems is proven under the given conditions.

Complete classification of invariant algebraic curves in the non-strongly minimal case.

Abstract

Consider some non-zero complex numbers $a_i, b_i, c_i, d_i$ with $1 \leq i \leq n$ and the associated classical Lotka-Volterra systems \[ \begin{cases} x' = a_i xy + b_i y \newline y' = c_i xy + d_i y \text{ .} \end{cases} \] We show that as long as $b_i \neq d_i$ for all $i$ and $\{ b_i, d_i\} \neq \{ b_j, d_j\}$ for $i \neq j$, any tuples $(x_1,y_1) , \cdots , (x_m,y_m)$ of pairwise distinct, non-degenerate solutions of these systems are algebraically independent over $\mathbb{C}$, meaning $\mathrm{trdeg}((x_1,y_1) , \cdots , (x_m,y_m)/\mathbb{C}) = 2m$. Our proof relies on extending recent work of Duan and Nagloo by showing strong minimality of these systems, as long as $b_i \neq d_i$. We also generalize a theorem of Brestovski which allows us to control algebraic relations using invariant volume forms. Finally, we completely classify all invariant algebraic curves in the non-strongly minimal, $b_i = d_i$ case by using machinery from geometric stability theory.