“complex-like” Analysis in O-minimal Structures

Theorem 1. Let K = 〈K,+, ·, . . .〉 be an expansion of K all of whose atomic relations are definable in R. If K is a proper expansion of 〈K,+, ·〉 then the field R is definable in K. By “proper expansion of 〈K,+, ·〉”, we mean that there is a definable set in K which is not definable in 〈K,+, ·〉 (even with parameters). In particular, the theorem implies that there are no proper expansions of an algebraically closed field which are stable and yet interpretable in an o-minimal structure. One can derive, for example, Chow’s classical theorem on analytic subsets of P(C) using this theorem (see [7, p. 340]). A weaker version of the theorem was proved earlier by D. Marker (see [4]), where the o-minimal structure R was assumed to be a real closed field. Here is another curious corollary, which was recently pointed out to us by M. Tressel. Consider an algebraically closed field K of characteristic zero and a set S ⊆ K. K may contain in general infinitely many non-isomorphic maximal real closed subfields. For any such field R, K can be identified with R, and hence S can be viewed as a subset of R. Let us call S o-minimal with respect to R if the structure 〈R,+, ·, S〉 is o-minimal (with the natural ordering of R). For example, every algebraic variety over K is o-minimal with respect to any maximal real closed subfield. This brings up an interesting question: