Strongly minimal expansions of (ℂ, +) definable in o-minimal fields

Abstract

We characterize those functions f:ℂ → ℂ definable in o‐minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o‐minimal field.

Strongly Minimal Reducts of Valued Fields

“…In this case, Proposition 3.20 need not hold, so it is natural to expect that Theorem 3.23 holds "up to (the action of) Aut K (K, +)". Note that this is exactly the case in [6], where a bi-interpretability result is obtained up to the action of GL 2 (R), and GL 2 (R) coincides with Aut R (C, +).…”

“…Let F = (F, + F , • F ) be a K-interpretable field given by Theorem 3.17. Our proof follows the lines of the proof of [6,Thm. 7.4].…”

“…In this subsection, we find a K-definable field. The proof follows the lines of the proof of[6, Thm 7.3].…”

“…In [7], Peterzil's conjecture was verified in the case of dim M (M) = 1 (showing that M is then locally modular). In an attempt to attack the case of dim M (M) = 2, Assaf Hasson and the first author (motivated by [15]) considered in [6] the case where the structure M expands (C, +) (or, more generally, the additive group of the algebraic closure of the underlying o-minimal field). By a general model-theoretic argument (see Proposition 3.2), in such a case Peterzil's conjecture reduces to the case of a strongly minimal structure of the form (C, +, X ), where X is an M-definable subset of C × C. It is shown in [6] (after quite a long argument) that if X is a graph of a function f, then f is an R-linear conjugate of a C-constructible function, which, in particular, verifies Peterzil's conjecture in this case.…”

Definable one dimensional structures in o-minimal theories

“…We deal with the special case where N is definable in M and dim M (N ) = 1 (see [9] and [20] for the investigation of certain cases of definable, stable, two-dimensional structures). We show:…”