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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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Quantifier Elimination

Quantifier Elimination

(p.678) Chapter Sixteen Quantifier Elimination
Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Princeton University Press

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 first, the models are the models of Tsuperscript nl with small derivation; in the second, the derivation is not small. One can move from models of the first completion to models of the second completion by compositional conjugation. The chapter begins with a discussion of extensions controlled by asymptotic couples and then shows the uniqueness-up-to-isomorphism of Newton-Liouville closures of ω‎-free H-fields. It then constructs a ω‎-free ΔΩ‎-field extension of K with a useful semiuniversal property. It also deduces Theorem 7 about quantifier elimination with various interesting consequences and concludes by specifying the language of ΔΩ‎-fields and demonstrating the elimination of quantifiers with applications.

Keywords:   quantifier elimination, Liouville closed H-field, small derivation, compositional conjugation, extension, asymptotic couple, Newton-Liouville closure

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