Algebraic independence of solutions to multiple Lotka-Volterra systems

Abstract

Consider some non-zero complex numbers $a_i,b_i,c_i,d_i$ with $1\le i\le n$ and the associated classical Lotka-Volterra systems \[ \begin{cases} x' = a_i xy + b_i x \newline y' = c_i xy + d_i y \text{ .} \end{cases} \] We show that as long as $b_i\ne d_i$ for all $i$ and $\{b_i,d_i\}\ne \{b_j,d_j\}$ for $i\ne j$, 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 $b_i\ne d_i$. 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.

Many improvements throughout the article