An intermediate value property for first-order differential polynomials (Q2758396)

This paper considers a Hardy field which is an ordered differential field of germs \(+\infty\) of differentiable real valued functions defined on real number half-lines \((a,\infty).\) Using the author's notion, where \(F\) denotes a differential polynomial over \(K,\) the following is one of the major results established. Let \(K\) be a Hardy field and \(F(Y,Z)\in K[Y,Z].\) Suppose that \(\theta, \psi \in K\) with \(\theta < \psi\) such that \(F(\theta,\theta')\) and \(F(\psi,\psi')\) are non-zero and of opposite sign in \(K.\) Then there exists an element \(\eta\) in a Hardy field extension of \(K\) such that \(\theta < \eta <\psi\) and \(F(\eta,\eta') = 0.\)NEWLINENEWLINEFor the entire collection see [Zbl 0971.00010].