When any four solutions are independent

TL;DR

The paper addresses when independence of solutions to a rational difference equation propagates from four generic solutions to all solutions. It translates the problem into the model theory of ACFA via rational $\sigma$-types, and uses pseudofinite permutation-group theory to bound the degree of nonminimality, establishing that if any four generic solutions are algebraically independent, then all are; in the autonomous case the degree of nonminimality is at most $2$, while in the nonautonomous case it is at most $\dim X+3$. The results interpret independence and invariant subvarieties in terms of $σ$-varieties, connecting algebraic dynamics with model-theoretic forking geometry. The work relies on a blend of dynamics, model theory, and finite-group classifications to constrain possible invariant configurations, with potential implications for Zariski-dense orbit phenomena and periodic subvarieties in algebraic dynamics.

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

Theorems & Definitions (31)

• Theorem A
• Theorem B
• Theorem C
• Remark 1.1
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 3.5
• Lemma 3.6