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

Abstract

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

11 pages

References in corpus (2)

• Distal and non-distal NIP theories
• Dimension, matroids, and dense pairs of first-order structures

Cited by in corpus (1)

• Monotone $T$-convex $T$-differential fields