# Hardy fields, the intermediate value property, and $\omega$-freeness

## TL;DR

This paper explores the properties of maximal Hardy fields, focusing on their intermediate value property for differential polynomials and their relation to transseries, establishing that all such fields are omega-free.

## Contribution

It proves that every maximal Hardy field is omega-free, advancing understanding of their structure and their connection to the intermediate value property.

## Key findings

- Maximal Hardy fields are omega-free.
- Equivalence between the intermediate value property and elementary equivalence to transseries.
- Supports the conjecture relating Hardy fields and transseries.

## Abstract

We discuss the conjecture that every maximal Hardy field has the Intermediate Value Property for differential polynomials, and its equivalence to the statement that all maximal Hardy field are elementarily equivalent to the differential field of transseries. As a modest but essential step towards establishing the conjecture we show that every maximal Hardy field is $\omega$-free.