Algebraic independence of the solutions of the classical Lotka-Volterra system

2025/02/24 by Duan, Yutong, Nagloo, Joel · 3 citations
#03C60 #12H05 #34M15 #Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2502.17194

Abstract

Let (x₁,y₁),…,(x_n,y_n) be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system \beginsplit x'amp;= axy + bx
y'amp;= cxy + dy, \endsplit where a,b,c,d∈ℂ∖\0\. We show that if (d)/(b)\not∈ℚ, then the solutions are algebraically independent over ℂ, that is tr.deg_ℂℂ(x₁,y₁,…,x_n,y_n)=2n. As a main part of the proof, we show that the set defined by the system in universal differential fields, with (d)/(b)\not∈ℚ, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general 2d-Lotka-Volterra system.

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