When any four solutions are independent

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 $\ddot{I}ƒ$-varieties. Surprisingly, the proofs of these results rely on the classification of finite simple groups.