Can we use Positivstellensatz (P-satz) below for a non-polynomial term?
P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the cone generated by $f$, $M$ the multiplicative monoid generated by $g$ and $I$ the ideal generated by $h$. Then the following properties are equivalent:
(i) The set $\{x\in R^n| f\geq 0, g\neq 0 , h=0\}$ is empty (ii) There exist $f \in P, g\in M, h\in I$ such that $f+g^2 +h=0$.
Reference: Bochnak et al. 1999