We give geometric axioms for existentially closed Hasse fields. We prove a quantifier elimination result for existentially closed n-truncated Hasse fields and characterize them as reducts of existentially closed Hasse fields.

Using methods of geometric stability , we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties.

We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their theory, using a sequence of e commuting Hasse derivations. In contrast to [6] our Hasse derivations are iterative.

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme.

It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected to Ax's theory of pseudofinite fields, and analogues are obtained for most of Ax's classical results. Some indication is given of the connection to nonstandard Frobenius maps.

Let T be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of T{“σ is an automorphism”} is an elementary class if and only if T has the definable multiplicity property, as long as T is a finite cover of a strongly minimal theory which does have the definable multiplicity property. We obtain cleaner results working with several automorphisms, and prove: the class of existentially closed models of T{“σi is an automorphism”: i=1,2} is (...) an elementary class if and only if T has the definable multiplicity property. (shrink)

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.