On a Differential Intermediate Value Property

TL;DR

This paper characterizes when Liouville closed H-fields possess the Intermediate Value Property for differential polynomials, linking this property to their elementary equivalence with the field of transseries, with implications for Hardy fields.

Contribution

It establishes a precise criterion for the Intermediate Value Property in Liouville closed H-fields, connecting it to their model-theoretic equivalence with transseries fields.

Findings

K has the Intermediate Value Property iff it is elementarily equivalent to transseries fields.

The result applies to Hardy fields.

Provides a model-theoretic characterization of differential fields.

Abstract

Liouville closed $H$-fields are ordered differential fields whose ordering and derivation interact in a natural way and where every linear differential equation of order $1$ has a nontrivial solution. (The introduction gives a precise definition.) For a Liouville closed $H$-field $K$ with small derivation we show: $K$ has the Intermediate Value Property for differential polynomials iff $K$ is elementarily equivalent to the ordered differential field of transseries. We also indicate how this applies to Hardy fields.