标题： 多个Lotka-Volterra系统的解的代数独立性 显示英文标题

标题： Algebraic independence of solutions to multiple Lotka-Volterra systems

摘要： 考虑一些非零复数$a_i, b_i, c_i, d_i$与$1 \leq i \leq n$相关的经典 Lotka-Volterra 系统\[ \begin{cases} x' = a_i xy + b_i y \newline y' = c_i xy + d_i y \text{ .} \end{cases} \]我们证明，只要对于所有$i$有$b_i \neq d_i$，并且对于$i \neq j$有$\{ b_i, d_i\} \neq \{ b_j, d_j\}$，这些系统的任何成对不同、非退化的解的元组$(x_1,y_1) , \cdots , (x_m,y_m)$在$\mathbb{C}$上代数独立，这意味着$\mathrm{trdeg}((x_1,y_1) , \cdots , (x_m,y_m)/\mathbb{C}) = 2m$。 我们的证明依赖于通过展示这些系统的强极小性来扩展Duan和Nagloo的近期工作，只要$b_i \neq d_i$。 我们还推广了Brestovski的一个定理，这使我们能够使用不变体积形式来控制代数关系。 最后，我们通过使用几何稳定性理论中的工具，完全分类了非强极小、$b_i = d_i$情况下的所有不变代数曲线。 显示英文摘要

摘要： 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.