The Positivstellensatz for definable functions on o-minimal structures

Abstract

In this note we prove two Positivstellensätze for definable functions of class $C^r$, $0\le r < \infty$, in any $o$-minimal structure $\mathcal{S}$ 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 $F=\{f_1\ge 0,\dots, f_k\ge 0\}$.