A transformal transcendence result for algebraic difference equations
Moshe Kamensky, Rahim Moosa
TL;DR
This work develops a model-theoretic framework for transformal transcendence in algebraic difference equations, proving that for equations of the form $\sigma^n(y)=f(y,\sigma(y),\dots,\sigma^{n-1}(y))$ over a characteristic-zero field with $\sigma$ acting on the base field, a nontrivial algebraic relation among pairwise $\sigma$-disjoint solutions (and their $\sigma$-transforms) collapses to a relation among as few as three such solutions; in the autonomous case a stronger $C_3$-to-$C_m$ bound holds for all $m$, while the general case yields a bound $C_{n+4}$. The paper develops a robust model-theoretic toolkit for rational dynamics: rational $\sigma$-varieties, quantifier-free types, finite binding groups, and canonical bases in the ACFA$_0$ setting. It proves key structure theorems, including a Zilber dichotomy for minimal rational types (one-based or finite-to-one map to a fixed-field-internal type), an exchange principle characterizing strong primitivity, and explicit bounds on the degree of nonminimality $\operatorname{nmdeg}(p)$ with sharp autonomous-case improvements. The disintegration theory links geometric simplicity of the base variety to transformal independence of multiple realizations, culminating in the main transformal transcendence result (Theorem c3thm-intro). The paper also provides concrete examples illustrating nonminimal types with exchange and 2-transitive binding-group actions, clarifying how nmdeg and disintegration interact with isotrivial dynamics and binding-group structure. Overall, it advances the understanding of how algebraic-dynamical properties of difference equations are reflected and constrained by model-theoretic types and their internality to the fixed field.
Abstract
Given an algebraic difference equation of the form \[σ^n(y)=f\big(y, σ(y),\dots,σ^{n-1}(y)\big)\] where $f$ is a rational function over a field $k$ of characteristic zero on which $σ$ acts trivially, it is shown that if there is a nontrivial algebraic relation amongst any number of $σ$-disjoint solutions, along with their $σ$-transforms, then there is already such a relation between three solutions. Here ``$σ$-disjoint" means $a\neqσ^r(b)$ for any integer $r$. A weaker version of the theorem, where ``three" is replaced by $n+4$, is also obtained when $σ$ acts non-trivially on $k$. Along the way a number of other structural results about primitive rational dynamical systems are established. These theorems are deduced as applications of a detailed model-theoretic study of finite-rank quantifier-free types in the theory of existentially closed difference fields of characteristic zero. In particular, it is also shown that the degree of non-minimality of such types over fixed-field parameters is bounded by $2$.