Defining integer valued functions in rings of continuous definable functions over a topological field

by L. Darnière and M. Tressl

Journal of Mathematical Logic 20, No. 03 (2020), 23 pages

Submitted on November 08, 2018

Abstract

Let K be an expansion of either an ordered field (K,≤), or a valued field (K,v). Given a definable subset X of K^m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension < 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions in C(X) which take values in Z is definable in C(X).

Mathematics Subject Classification

03C64 Model theory of ordered structures; o-minimality
12J10 Valued fields
12L05 Decidability [See also 03B25]

Electronic version of the paper

Version October 29, 2018 (24 pages) pdf