Constructing $\omega$-free Hardy fields

TL;DR

This paper proves that every Hardy field can be extended to an omega-free Hardy field, connecting classical oscillation criteria with new extensions, and addressing questions posed by Boshernitzan.

Contribution

It introduces a method to extend any Hardy field to an omega-free Hardy field, advancing the understanding of their structure and applications.

Findings

Every Hardy field extends to an omega-free Hardy field.

The result links oscillation criteria with Hardy field extensions.

Applications include answering questions of Boshernitzan.

Abstract

We show that every Hardy field extends to an $\omega$-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [10], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.