When any four solutions are independent
Logic
2025-10-23 v3 Algebraic Geometry
Dynamical Systems
Abstract
We show that if any four distinct solutions of a rational difference equation are algebraically independent, then any number of distinct solutions to the equation are independent. A nontrivial variant of this result is given for autonomous difference equations or algebraic dynamical systems, where we show the degree of nonminimality is at most one. The results have a natural interpretation in terms of invariant or periodic subvarieties of algebraic dynamical systems and -varieties. Surprisingly, the proofs of these results rely on the classification of finite simple groups.
Cite
@article{arxiv.2510.08387,
title = {When any four solutions are independent},
author = {James Freitag},
journal= {arXiv preprint arXiv:2510.08387},
year = {2025}
}