The strict Positivstellensatz for global analytic functions and the moment problem for semianalytic sets

Abstract

Analytic functions strictly positive on a global semianalytic set X = {f1 0, · · · , fk 0} in Rn are characterized as functions expressible as g = a0+a1f1+· · ·+akfk for strictly positive global analytic functions a0, · · · , ak. The proof is elementary, using the fact that the analytic functions are dense in C(Rn,R) in the Whitney topology. The same proof works for Nash functions. This is an improvement of the standard analytic version of Stengle’s Positivstellensatz in two directions: The hypothesis is weaker (there is no requirement that X be compact) and the conclusion is stronger. Several applications are given including: (i) a new proof of the weak Positivstellensatz for semianalytic sets; and (ii) the solution of theK-moment problem for basic closed semianalytic