A dichotomy for T-convex fields with a monomial group

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.

Automated summary

This study proves a dichotomy for o-minimal fields R expanded by a T-convex valuation ring and a compatible monomial group. It demonstrates that if T is power bounded, then this expansion of R is model complete, has a distal theory, and the definable sets are geometrically tame. However, if R defines an exponential function, then the natural numbers can be externally definable in our expansion, precluding any sort of model-theoretic tameness.