Submission date: 18 December 2020

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.

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Full text arXiv 2012.09993: pdf, ps.