Let M be a C-minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → T in C-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a de...

An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed graph specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. In this paper, we consider the definability of functions by means of arithmetic circuits. We prove two negative results: the first shows, roughly, that a function is not circuit-definable if it has an infinite r...

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of Rn is the restriction of a Cm-function on Rn. A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the C1-case of Whitney’s Extension Problem in this context. Our main tool is a defina...

We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a function f : [0,∞) → R is such that f(N) ⊆ Z, then either sup|x̄|≤r f(x̄) grows faster than exp(r), for some δ > 0,...

We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably h...