On a Differential Intermediate Value Property

Matthias Aschenbrenner; Lou van den Dries; Joris van der Hoeven

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arXiv · 2105.12631

On a Differential Intermediate Value Property

Matthias Aschenbrenner·Lou van den Dries·Joris van der Hoeven

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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.

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Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven. 2021-05-26. On a Differential Intermediate Value Property. https://arxiv.org/abs/2105.12631

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