The Positivstellensatz for definable functions on O-minimal structures

In this note we prove two Positivstellensatze for definable functions of class C-r, 0 less than or equal to r < ∞, in any o-minimal structure S expanding a real closed field R. Namely, we characterize the definable functions that are nonnegative (resp. strictly positive) on basic definable sets of the form F = {f(1) ≥ 0,...,f(k) ≥ 0}.