A dichotomy for T $T$ -convex fields with a monomial group

Abstract

We prove a dichotomy for o-minimal fields $R$, expanded by a $T$-convex valuation ring (where $T$ is the theory of $R$) and a compatible monomial group. We show that if $T$ is power bounded, then this expansion of $R$ is model complete (assuming that $T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if $R$ defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model-theoretic tameness.