#### Abstract for the paper

## Exponentiation in power series fields

* by Franz-Viktor Kuhlmann, Salma Kuhlmann and Saharon
Shelah*

We prove that for no nontrivial ordered abelian group G, the ordered
power series field **R**((G)) over the reals **R** admits an
exponential, i.e., an isomorphism between its ordered additive group and
its ordered multiplicative group of positive elements, but that there is
a non-surjective logarithm. For an arbitrary ordered field k, no
exponential on k((G)) is compatible, that is, induces an exponential
on k through the residue map. This is proved by showing that certain
functional equations for lexicographic powers of ordered sets are not
solvable.

*Last update: February 3, 1999*