Comparison of Exponential-Logarithmic and Logarithmic-Exponential series

Marcus Tressl, Salma Kuhlmann

Published 2011-12-18Version 1

We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} and \cite {KS}. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th$(\R_{an, exp})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.