Short Hardy fields

TL;DR

This paper proves that differential algebraic extensions of short Hardy fields remain short within the broader context of H-fields, and demonstrates an isomorphism between certain asymptotic couples and analytic Hardy fields.

Contribution

It establishes that Hardy field extensions preserve shortness in H-fields and extends Rosenlicht's theorem to relate asymptotic couples with analytic Hardy fields.

Findings

Differential algebraic Hardy field extensions of short Hardy fields are short.

Every short asymptotic couple of Hardy type with small derivation is isomorphic to an analytic Hardy field's asymptotic couple.

The results generalize properties of Hardy fields within the framework of H-fields.

Abstract

Differentially algebraic Hardy field extensions of short Hardy fields are short. This is proved in the more general setting of $H$-fields. As an application we extend a theorem of Rosenlicht (1981) by showing that each short asymptotic couple of Hardy type with small derivation is isomorphic to the asymptotic couple of an analytic Hardy field.