An Intermediate Value Property for First-Order Differential Polynomials

In the last century Du Bois-Reymond promoted the idea that the archimedean real continuum should be viewed as only a small part of a larger nonarchimedean realm of orders of infinity. This notion finds an elegant and efficient expression in the theory of Hardy fields.

A natural question is whether the properties of the real line that make it an {\em ordered continuum\/} persist in this nonarchimedean realm, where differentiation is continuous. This line of thought suggests the following problem:

Let $F(Y,Y',\dots,Y^{(n)})\in K[Y,Y',\dots,Y^{(n)}]$ be a differential polynomial over a Hardy field $K$. Let $\phi < \psi$ in $K$ such that $F(\phi, \phi',\dots, \phi^{(n)})$ and $F(\psi, \psi',\dots, \psi^{(n)})$ are non-zero and of opposite sign in $K$. Does it follow that there exists an element $\eta$ in a Hardy field extension of $K$ such that $\phi < \eta < \psi$ and $F(\eta, \eta',\dots,\eta^{(n)})=0$?

For $n=0$ we have a well-known affirmative answer due to A. Robinson. We obtain an affirmative answer for $n=1$.