A model theoretic application of Gelfond-Schneider theorem

We prove that weakly o-minimal expansions of the ordered field of all real algebraic numbers are polynomially bounded. Apart of this we make a couple of observations concerning weakly o-minimal expansions of ordered fields of finite transcendence degree over the rationals. We show for instance that if Schanuel’s conjecture is true and K ⊆ R is a field of finite transcendence degree over the rationals, then weakly o-minimal expansions of (K,≤,+, ·) are polynomially bounded. 0 Introduction G. Faber in [Fa] (see also [Ma], Chapter II, §36) gives a construction of an entire transcendental function f(z) = ∞ ∑ n=0 anz n with rational coefficients an such that all its values along with all derivatives are algebraic numbers at all algebraic arguments. Thus, one cannot detect the transcendence of f by examining the values assumed by f or its derivatives at algebraic arguments. An extreme form of such a behavior in the real case was discovered in 1995 by A. Wilkie (see [Wi]). He described a construction of an everywhere analytic transcendental function g : R −→ R, whose transcendence cannot be detected by any first order methods. In other words, the ordered field of all real algebraic numbers Ralg := (Ralg,≤,+, ·) when expanded by g 1 Ralg is an elementary substructure of (R,≤, +, ·, g). The Wilkie’s construction, thanks to results of [DD], provides an example of a proper o-minimal expansion of Ralg, answering positively a question asked in [LS] (see the last paragraph of §5). As Wilkie’s o-minimal expansion of Ralg turns out to be polynomially bounded, and there are several known o-minimal expansions of R := (R,≤, +, ·) which are not polynomially bounded, it is natural to ask about the existence of o-minimal expansions of Ralg which are not polynomially bounded. This paper gives a negative answer to that question, even if one relaxes the hypothesis of o-minimality to that of weak o-minimality. The paper is organized as follows. In §1 we recall some basic notation and terminology concerning o-minimality and weak o-minimality, paying special attention to so called weakly o-minimal non-valuational expansions of ordered groups. We skip the most complicated inductive definitions referring the reader to [We07], where the basic model theory of weakly o-minimal non-valuational expansions of ordered groups is developed. In §2 we prove the main result of the paper (Theorem 2.2). It says that all weakly o-minimal structures expanding Ralg are polynomially bounded. The proof combines the author’s work on weakly o-minimal non-valuational expansions of ordered groups with Baizhanov’s results concerning expansions of models of weakly o-minimal theories by families of convex predicates, Miller’s dichotomy for o-minimal expansions of R, and the Gelfond-Schneider theorem. 1This research was supported by a Marie Curie Intra-European Fellowships within the 6th European Community Framework Programme. Contract number: MEIF-CT-2003-501326.