Constant power maps on Hardy fields and transseries

TL;DR

This paper proves that expanding the differential field of transseries with a power map is model complete, providing an effective axiomatization and establishing a transfer theorem between Hardy fields and transseries.

Contribution

It demonstrates model completeness of the transseries expansion with power maps and provides an effective axiomatization, linking Hardy fields and transseries.

Findings

The expansion is model complete.

An effective axiomatization is provided.

A transfer theorem between Hardy fields and transseries is established.

Abstract

Let $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We consider the expansion of $\mathbb{T}$ by the binary map that sends a real number $r$ and a positive transseries $f$ to the transseries $f^r$. Building on recent work of Aschenbrenner, van den Dries, and van der Hoeven, we show that this expansion is model complete, and we give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. We show that maximal Hardy fields, equipped with the same map $(f,r)\mapsto f^r$, enjoy the same theory as $\mathbb{T}$, and we use this to establish a transfer theorem between Hardy fields and transseries.