Algebraic independence of solutions to multiple Lotka-Volterra systems

2025/07/23 by Duan, Yutong, Eagles, Christine, Jimenez, Léo · 1 citation
#03C69 #12H05 #34M15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2507.17090

Abstract

Consider some non-zero complex numbers a_i, b_i, c_i, d_i with 1 ≤ i ≤ n and the associated classical Lotka-Volterra systems \begincases x' = a_i xy + b_i y \newline y' = c_i xy + d_i y . \endcases We show that as long as b_i ≠ d_i for all i and \ b_i, d_i\ ≠ \ b_j, d_j\ for i ≠ j, any tuples (x₁,y₁) , ⋯ , (x_m,y_m) of pairwise distinct, non-degenerate solutions of these systems are algebraically independent over ℂ, meaning trdeg((x₁,y₁) , ⋯ , (x_m,y_m)/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 ≠ 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.

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Cited by

• Integration in finite terms and exponentially algebraic functions

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