Algebraic independence of solutions to multiple Lotka-Volterra systems
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909706099884032 |
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| author | Duan, Yutong Eagles, Christine Jimenez, Léo |
| author_facet | Duan, Yutong Eagles, Christine Jimenez, Léo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic independence of solutions to multiple Lotka-Volterra systems Duan, Yutong Eagles, Christine Jimenez, Léo Logic Classical Analysis and ODEs 34M15, 12H05, 03C69 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. |
| title | Algebraic independence of solutions to multiple Lotka-Volterra systems |
| topic | Logic Classical Analysis and ODEs 34M15, 12H05, 03C69 |
| url | https://arxiv.org/abs/2507.17090 |