0 Fields with pseudo - exponentiation

1 Introduction This research is motivated by the study of model-theoretical properties of classical 'analytic' structures, i.e. ones having natural analytic representation (see also [Z]). For example, the structure of complex numbers as a field with exponentiation C exp = (C, +, ·, exp). One of the questions we can ask is whether C exp is quasi-minimal, i.e. any definable subset of C exp is either countable or of power continuum. Another question is about homogeneity of the structure; we do not know any its automorphism except the identity and the complex conjugation. In general we would like to understand the nature of analytic dimension in a context close to model-theoretic stability theory. A slightly weaker analytic structure C (2) exp is a two-sorted structure with both sorts C(1) and C(2) being copies of complex numbers, on both sorts the field structure is given and there is a mapping exp : C(1) → C(2) in the language. The model theory of the both structures, as well as of many others of this kind, seems very hard to study directly. We study here model theory of abstract analogues of C (2) exp. We start by considering the class E p of two-sorted structures of the form (D, ex, R), D, the domain of a mapping ex is a field of characteristic zero, R, a field of characteristic p and ex is a homomorphism of the additive group of D onto the multiplicative group R × of R. Following the ideology of Hrushovski's construction of non-classical structures with nice dimension notion (see [H]), 1 we consider a notion of a predimension δ(X) for finite subsets X ⊆ D defined as δ(X) = tr.d.(X) + tr.d.(ex(X)) − dim Q (X), here dim Q (X) is the dimension of the Q-linear space generated by X. We then define the subclass S p of E p , the weak Schanuel class as the class of structures where δ(X) ≥ 0 for all X and the kernel of ex is standard (a cyclic additive subgroup in case p = 0). Recall that the Schanuel conjecture [L] states that, given Q-linearly independent complex numbers x holds. We prove that the weak Schanuel class S p is non-empty. We study the predimension δ and the relation A ≤ B following the Hrushovski pattern. We define a notion of exponentially-algebraically closed structures, which are just existentially closed structures with …