Model completeness of o-minimal fields with convex valuations

Abstract

We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).