Valuation theory of exponential Hardy fields I

In this paper, we analyze the structure of the Hardy fields associated with o-minimal expansions of the reals with exponential function. In fact, we work in the following more general setting. We take T to be the theory of a polynomially bounded o-minimal expansion P of the ordered field of real numbers. By FT we denote the set of all 0-definable functions of P . Further, we assume that T defines the restricted exponential and logarithmic functions (cf. [D–M–M1]). Then also T (exp) is o-minimal (cf. [D–S2]). Here, T (exp) denotes the theory of the expansion (P , exp) where exp is the un-restricted real exponential function. Finally, we take any model R of T (exp) which contains (R ,+, ·, <,FT , exp) as a substructure. Then we consider the Hardy field H(R) (see Section 2.2 for the definition) as a field equipped with convex valuations. Theorem B of [D–S2] tells us that T (exp) admits quantifier elimination and a universal axiomatization in the language augmented by a symbol for the inverse function log of exp. This implies that H(R) is equal to the closure of its subfield R(x) under FT , exp and log; here, x denotes the germ of the identity function (cf. [D–M–M1], §5; the arguments also hold in the case where R is a non-archimedean model). We shall analyze the valuation theoretical structure of this closure by explicitly showing how it can be built up from R(x) (cf. Section 3.3). Our construction method yields the following result (see Section 3.4 for definitions):