A transformal transcendence result for algebraic difference equations

Abstract

Given an algebraic difference equation of the form \[\sigma^n(y)=f\big(y, \sigma(y),\dots,\sigma^{n-1}(y)\big)\] where $f$ is a rational function over a field $k$ of characteristic zero on which $\sigma$ acts trivially, it is shown that if there is a nontrivial algebraic relation amongst any number of $\sigma$-disjoint solutions, along with their $\sigma$-transforms, then there is already such a relation between three solutions. Here ``$\sigma$-disjoint" means $a\neq\sigma^r(b)$ for any integer $r$. A weaker version of the theorem, where ``three" is replaced by $n+4$, is also obtained when $\sigma$ 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 $2$.