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Definable functions in tame expansions of algebraically closed valued fields. (English) Zbl 1471.03066

It was shown in 1990’s that:
– the real field expanded by the exponential function is o-minimal [A. J. Wilkie, J. Am. Math. Soc. 9, No. 4, 1051–1094 (1996; Zbl 0892.03013)];
– an o-minimal expansion of the real field is either polynomially bounded or the exponential function is definable in it [C. Miller, Proc. Am. Math. Soc. 122, No. 1, 257–259 (1994; Zbl 0808.03022)].
In the paper under review, the authors consider the theory ACVF (algebraically closed valued fields) and C-minimality is taken as the tameness condition replacing o-minimality. The authors show two types of results regarding definable functions in the set-up described above:
– “type (I)” results about behaviour around infinity;
– “type (II)” results about local behaviour at almost all points of the domain.
A very interesting application of type (I) results is obtained: C-minimal models of ACVF with the smallest possible value group (being the rational numbers) are polynomially bounded. Therefore, the exponential case of the o-minimal dichotomy does not occur in such a valued context.
It is natural to ask whether the condition on the value group is necessary here. The authors address this issue by asking (Question 1) whether all C-minimal valued fields are polynomially bounded. The type (II) results are used to show some kind of factorizations of definable functions on the RV-sort.

MSC:

03C64 Model theory of ordered structures; o-minimality
03C60 Model-theoretic algebra
12L12 Model theory of fields
12J10 Valued fields
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References:

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