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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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Newtonianity of Directed Unions

Newtonianity of Directed Unions

Chapter:
(p.671) Chapter Fifteen Newtonianity of Directed Unions
Source:
Asymptotic Differential Algebra and Model Theory of Transseries
Author(s):

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175423.003.0016

This chapter considers the newtonianity of directed unions and proves an analogue of Hensel's Lemma for ω‎-free differential-valued fields of H-type: Theorem 15.0.1. Here K is an H-asymptotic field with asymptotic couple (Γ‎, ψ‎), and γ‎ ranges over Γ‎. The chapter first describes finitely many exceptional values, integration and the extension K(x), and approximating zeros of differential polynomials before proving Theorem 15.0.1, which states: If K is d-valued with ∂K = K, and K is a directed union of spherically complete grounded d-valued subfields, then K is newtonian. In concrete cases the hypothesis K = ∂K in the theorem can often be verified by means of Corollary 15.2.4.

Keywords:   newtonianity, directed union, Hensel's Lemma, differential-valued field, H-asymptotic field, asymptotic couple, exceptional value, integration, extension, differential polynomial

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