What does variety language mean?

Question

In Zilber's paper entitled 'a theory of generic function with derivations', he works in the variety language for fields? What does this mean? If I want to define the variety language what can I say?

Answer 1

(I leave this reply just in case this is still relevant)

Let $V$ be a varity (can be any definable set actually) in an algebraically closed field $k$. For any subset of $V^n, n \geq 1$ relatively closed in $V^n$ in Zariski topology introduce a predicate --- this is what is called the "variety language". Definable sets in this language are the same as definable subsets of $V^n$ (by quantifier elimination in $ACF_p$ and stable embeddedness of $V$).