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Let $(K,\sigma)$ be a difference field of characteristic $0$, i.e. equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $\delta=0$ for some $\delta\in K\{x\}$, where $$K\{x\}=K\left[\sigma^i(x):i\in\mathbb N\right].$$ Let $F=\{x\in K:\sigma x=x\}$ be the subfield fixed by $\sigma$.

Question. Is $[K:F]$ finite?

Remark. The answer is yes if the identity is of 'degree 1', i.e. of the form $$\sum_{k=0}^n a_k\sigma^k(x)=a_{{-1}}$$ for some $a_{-1},a_0,\dots,a_n\in K$, and conversely, a finite dimensional $F$-algebra satisfies such an identity.

Edit. The following question is motivated by the question:

Question. Is the evaluation map that maps a difference polynomial $\delta\in K\{x\}$ to the function $\delta_K:K\rightarrow K$ is injective, at least when $[K:F]$ is infinite?

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The answer is yes and appears in R. Cohn's book 'Difference algebra' Lemma II p. 201.

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