Valuation theory of exponential Hardy fields II: Principal parts of germs in the Hardy field of o-minimal exponential expansions of the reals

Abstract

We present a general structure theorem for the Hardy field of an o-minimal expansion of the reals by restricted analytic functions and an unrestricted exponential. We proceed to analyze its residue fields with respect to arbitrary convex valuations, and deduce a power series expansion of exponential germs. We apply these results to cast "Hardy's conjecture" (see \cite[p.111]{[KS]}) in a more general framework. This paper is a follow up to \cite{[K-K2]} and is partially based on unpublished results of \cite{[K-K]}. A previous version \cite{[K-K1]} (which was dedicated to Murray A. Marshall on his 60th birthday) remained unpublished. In \cite{[W]} our structure theorem for the residue fields was rediscovered and applied to the diophantine context. Due to this revived interest, we decided to rework the preprint \cite{[K-K1]} and submit it to the Proceedings Volume.

arXiv comment15 pages, to appear in AMS Contemporary Mathematics (CONM); Proceedings of the Conference on Ordered Algebraic Structures and Related Topics. Edited by: F. Broglia, F. Delon, M. Dickmann, D. Gondard, and V. Powers (2016)