Revista de la
Unión Matemática Argentina
On a differential intermediate value property
Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven
Volume 64, no. 1 (2022), pp. 1–10

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Liouville closed $H$-fields are ordered differential fields where the ordering and derivation interact in a natural way and 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 that $K$ has the Intermediate Value Property for differential polynomials if and only if $K$ is elementarily equivalent to the ordered differential field of transseries. We also indicate how this applies to Hardy fields.