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

TL;DR

This paper develops a structure theorem for the Hardy field in o-minimal exponential expansions of the reals, analyzes its residue fields, and applies these to a generalized Hardy's conjecture, advancing valuation and power series expansion theories.

Contribution

It introduces a comprehensive structure theorem for Hardy fields in o-minimal settings and extends the analysis of residue fields and exponential germs, building on previous unpublished results.

Findings

Established a general structure theorem for Hardy fields in o-minimal exponential expansions.

Derived power series expansions of exponential germs.

Connected the results to Hardy's conjecture and valuation theory.

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 This paper is a follow up to \cite{[K-K2]} and is partially based on unpublished results of \cite{[K-K]}. A previous version \cite{[K-K1]} (which was dedicated to Murray A. Marshall on his 60th birthday) remained unpublished.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.