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Asymptotic Differential Algebra and Model Theory of Transseries$
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Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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Newtonian Differential Fields

Newtonian Differential Fields

(p.640) Chapter Fourteen Newtonian Differential Fields
Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Princeton University Press

This chapter deals with Newtonian differential fields. Here K is an ungrounded H-asymptotic field with Γ‎ := v(Ksuperscript 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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