S.: Exponential-logarithmic power series

In [K–K–S] it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows to construct for every κ regular uncountable cardinal, 2κ pairwise non-isomorphic models of real exponenti-ation (of cardinality κ), but all isomorphic as ordered fields. Indeed, the 2κ exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms. 1 Introduction. In [T], Tarski proved his celebrated result that the elementary theory of the ordered field of real numbers admits elimination of quantifiers, and gave a recursive axiom-atization of its class of models (the class of real closed fields). He asked whether analogous results hold for the elementary theory Texp of (R, exp) (the ordered fiel