# Newtonian Differential Fields

# Newtonian Differential Fields

This chapter deals with Newtonian differential fields. Here *K* is an ungrounded *H*-asymptotic field with Γ := *v*(*K*superscript x ) not equal to {0}. So the subset ψ of Γ is nonempty and has no largest element, and thus *K* is pre-differential-valued by Corollary 10.1.3. An extension of *K* means an *H*-asymptotic field extension of *K*. The chapter first considers the relation of Newtonian differential fields to differential-henselianity before discussing weak forms of newtonianity and differential polynomials of low complexity. It then proves newtonian versions of d-henselian results in Chapter 7, leading to the following analogue of Theorem 7.0.1: If *K* is λ-free and asymptotically d-algebraically maximal, then *K* is ω-free and newtonian. Finally, it describes unravelers and newtonization.

*Keywords:*
newtonization, Newtonian differential field, H-asymptotic field, differential-henselianity, newtonianity, differential polynomial, low complexity, unraveler

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