When any four solutions are independent

James Freitag

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arXiv · 2510.08387

When any four solutions are independent

James Freitag

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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 $\sigma$-varieties. Surprisingly, the proofs of these results rely on the classification of finite simple groups.

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James Freitag. 2025-10-09. When any four solutions are independent. https://arxiv.org/abs/2510.08387

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