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Asymptotic Fields and Asymptotic Couples

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0010
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with asymptotic differential fields and their asymptotic couples. Asymptotic fields include Rosenlicht's differential-valued fields and share many of their basic properties. A key ... More


Valued Differential Fields

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0007
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous P ∈ K K{Y}superscript Not ... More


Quantifier Elimination

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0017
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter considers the theory Tsuperscript nl of ω‎-free newtonian Liouville closed H-fields that eliminates quantifiers in a certain natural language. This theory has two completions: in the ... More


Newtonianity of Directed Unions

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0016
Subject:
Mathematics, Computational Mathematics / Optimization

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 ... More


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