I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) algebra being a field $\mathbb{F}$ and operations being polynomials and field automorphisms) I am also interested in computational aspects.


I am mainly interested in case of field of multivariate rational functions, i.e. $\mathbb{F} = K(x,y)$.
An algebraic set in such setting could be $$\{(f,g) \in (K(x,y))^2 \mid x^2-y^2-f \cdot g(x=\frac{x+y}{2}, y = \frac{x-y}{2}) = 0\}.$$ and ideals are additionally closed under field automorphisms.


I would be grateful for reference to any source that considers theory like this, also the computational aspects.



Algebraic geometry additionally equipped with field automorphism operation
You should look into [difference algebra](https://en.wikipedia.org/wiki/Difference_algebra), which is exactly this setting. (It's called difference algebra because the most classical version looked at the automorphism on polynomials that sends $f(x)$ to $f(x+1)$, which can be used to state the theory of difference equations.)


The hottest area in this is the study of the model theory of fields with automorphisms. Zoe Chatzidakis has some [lecture notes](https://www.math.ens.fr/%7Ezchatzid/papiers/Orsay-ZC.pdf) on the subject.


