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.
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