$T$-convex valued fields with tempered exponentiation

TL;DR

This paper develops a new theory for $T$-convex valued fields with a tempered exponential function, extending the structure and properties of power-bounded $T$-convex fields and enabling quantifier elimination and other model-theoretic features.

Contribution

It introduces the universal theory TKVF with a tempered exponential, bridging power-bounded TCVF and exponential TCVF, and establishes its well-behaved model-theoretic properties.

Findings

Quantifier elimination in a natural language

Definition of a generalized Euler characteristic

A notion of dimension for the theory

Abstract

We continue the effort of grokking the structure of power-bounded $T$-convex valued fields, whose theory is in general referred to as TCVF. In the present paper our focus is on certain expansion of it that is equipped with a tempered exponential function beyond the valuation ring. In order to construct such a tempered exponential function, the signed value group is also converted into a model of $T$ plus exponentiation and is in fact identified with (a section of) the residue field via the composition of a diagonal cross-section and an angular component map. In a sense, the resulting universal theory TKVF is a halfway point between power-bounded TCVF and exponential TCVF. This theory is reasonably well-behaved. In particular, we show that it admits quantifier elimination in a natural language, a notion of dimension, a generalized Euler characteristic, etc.