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I am currently reading David Marker's paper "A Remark on Zilber's Pseudoexponentiation" (J. Symb. Logic 71 (2006), no. 3). He describes the axioms for Zilber's pseudoexponential fields. The Strong Exponential Closure axiom is stated as follows:

For all finite $A$ if $V\subseteq G_n(K)$ is irreducible, free and normal there is $(\bar{x},E(\bar{x}))\in V$ a generic point of $V$ over $A$.

$A$ is any finite subset of $K$.

$G_n(K)=K\times K^{\times}$ where $K$ is the field, and $E$ is the exponential function on the field.

My question is what it means that a point is generic over a set. It seems to be an expression from algebraic geometry, which I don't know much about. I have already found out what a generic point is (using the Zariski Topology), but cannot find anywhere what it means for a point to be generic over a set.

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  • $\begingroup$ What is $A$ here? $\endgroup$
    – Hoot
    Commented Dec 8, 2014 at 13:20
  • $\begingroup$ Sorry, I fogot to mention that. $A$ seems to denote a finite subset of $K$. $\endgroup$
    – KDuck
    Commented Dec 8, 2014 at 13:29
  • $\begingroup$ Actually, I tracked down the paper and earlier on the page where the strong exponential closure axiom is stated, $A$ is set to be a matrix of rank $k$ for $1\leq k \leq n$ where $K$ is of transcendence degree at least $k$ over $\mathbb{Q}$. It doesn't look like the definition of $A$ changes between there and the statement of the axiom. $\endgroup$
    – Nick
    Commented Dec 8, 2014 at 19:01
  • $\begingroup$ That is very confusing in this paper. The choice of variables is poor. The next axiom (countable closure) states something about the definable closure of $A$. The closure is something defined for subsets of $K$. On the next page (Theorem 1.5) this is mentioned again. $\endgroup$
    – KDuck
    Commented Dec 9, 2014 at 9:27
  • $\begingroup$ Yeah, that's why I would recommend you read the other paper I sent you the link for first, or at least the overview of Zilber fields contained in it. $\endgroup$
    – Nick
    Commented Dec 9, 2014 at 16:22

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The overview of Zilber fields given in this paper adequately explains what the "generic point" language in that axiom means. The term is used in the sense of algebraic geometry but it requires quite a few preliminary definitions to unpack.

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