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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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Valued Differential Fields

Valued Differential Fields

(p.310) Chapter Six Valued 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 valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous PK K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.

Keywords:   valued differential field, differential field extension, residue field, small derivation, value group, asymptotic couple, dominant part, Equalizer Theorem, pseudocauchy sequence, immediate extension

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