Value Groups and Residue Fields of Models of Real Exponentiation

Abstract

Let $F$ be an archimedean field, $G$ a divisible ordered abelian group and $h$ a group exponential on $G$. A triple $(F,G,h)$ is realised in a non-archimedean exponential field $(K,\exp)$ if the residue field of $K$ under the natural valuation is $F$ and the induced exponential group of $(K,\exp)$ is $(G,h)$. We give a full characterisation of all triples $(F,G,h)$ which can be realised in a model of real exponentiation in the following two cases: i) $G$ is countable. ii) $G$ is of cardinality $\kappa$ and $\kappa$-saturated for an uncountable regular cardinal $\kappa$ with $\kappa^{<\kappa} = \kappa$. Moreover, we show that for any o-minimal exponential field $(K, \exp)$ satisfying the differential equation $\exp' = \exp$, its residue exponential field is a model of real exponentiation.