### Exponentiation in power series fields

by Kuhlmann and Kuhlmann and Shelah. [KKSh:601]

Proc American Math Soc, 1997

We prove that for no nontrivial ordered abelian group G, the
ordered power series field R((G)) 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.

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