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Exponentiation in power series fields

Authors: Franz-Viktor Kuhlmann, Salma Kuhlmann and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 125 (1997), 3177-3183
MSC (1991): Primary 12J15, 06A05; Secondary 12J25, 06F20
MathSciNet review: 1402868
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Abstract: We prove that for no nontrivial ordered abelian group $G$ does the ordered power series field ${\Bbb R}((G))$ admit 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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Additional Information

Franz-Viktor Kuhlmann
Affiliation: Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany

Salma Kuhlmann
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

Saharon Shelah

Keywords: Ordered exponential fields, power series fields, lexicographic products, convex valuations
Received by editor(s): January 31, 1996
Received by editor(s) in revised form: May 18, 1996
Additional Notes: The second author was supported by a Deutsche Forschungsgemeinschaft fellowship. The third author was partially supported by the Edmund Landau Center for research in Mathematical Analysis, and supported by the Minerva Foundation (Germany). Publication number 601.
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1997 American Mathematical Society